Séminaire Lotharingien de Combinatoire 62 (2010), Article B62d

ON THE COMBINATORICS OF YOUNG–CAPELLI SYMMETRIZERS

Francesco Regonati

Dipartimento di Matematica “Alma Mater Studiorum” Università degli Studi di Bologna

Abstract. We deal with the characteristic zero theory of supersymmetric algebras, regarded as bimodules under the action of a pair of general linear Lie superalgebras, as developed by Brini et al. (see [Proc. Natl. Acad. Sci. USA 85 (1988), 1330–1333; Proc. Natl. Acad. Sci. USA 86 (1989), 775–778] for the first version, [Séminaire Lotharingien Combin. 55 (2007), Article B55g, 117 pp.] for the last version; see also Sergeev [Mat. Sb. (N.S.) 123(165) (1984), 422–430; Michigan Math. J. 49 (2001), 113–146] and Cheng and Wang [Compositio Math. 128 (2001), 55–94]). The theory had its roots in the pioneering work of Grosshans, Rota and Stein [Invariant theory and Superalgebras, Amer. Math. Soc., Providence, RI, 1987], Berele and Regev [Bull. Am. Math. Soc. 8 (1983), 337–339; Adv. Math. 64 (1987), 118–175] and Sergeev [Mat. Sb. (N.S.) 123(165) (1984), 422–430]. The basic objects of the theory, i.e., symmetrized bitableaux and Young–Capelli symmetrizers, are defined by means of a superalgebraic extension of Capelli’s method of virtual variables, and the relations between them are proved in the virtual setting, by means of a Triangularity Lemma, a Nondegeneracy Lemma, and the Superstraightening Law. We give a detailed exposition of the foundations of this theory. In doing this, we establish three new propositions on virtual expressions, and give new, elementary combinatorial proofs of the Triangularity Lemma and of the Nondegeneracy Lemma. With these proofs, we complete the process of giving the theory an elementary combinatorial foundation.

Introduction

We deal with the characteristic zero theory of supersymmetric algebras, regarded as bimodules under the action of a pair of general linear Lie superalgebras, as developed by Brini et al. (see [8, 9] for the first version, [6] for the last version; see [10, 11, 7] for ramifications in other directions; see also [24]). For each module, the theory yields a basis, the basis of standard symmetrized bitableaux, for the space, and a basis, the basis of standard Young–Capelli symmetrizers, for the operator algebra induced by the action, which acts on the former basis in a nondegenerate triangular way. These bases are used to get explicit complete decompositions of the space as a semisimple module, and of the operator algebra as a semisimple algebra. The basic objects are defined by means of a superalgebraic version of Capelli’s method of virtual variables, and the relations between them are proved in the virtual setting. Roughly speaking, the foundations of this theory in the first version relied on Grosshans, Rota and Stein’s Superstraightening Law [26], and on Berele and Regev’s supertensor representation theory of general linear Lie superalgebras and symmetric groups ([1, 2]; see also [35]).

In the last version, Berele and Regev’s theory is replaced by the ordinary representation theory of the symmetric group. A first use of Capelli’s method of virtual variables in a superalgebraic context can be traced back to the pioneering work of Koszul [28]. The idea of virtual variables is the basic tool introduced by Capelli in order to prove his famous identities [15].

The theory we deal with yields, by specialization, several classical theories. We mention: the ordinary representation theory of the symmetric group, in which symmetrized bitableaux and Young–Capelli symmetrizers turn out to be the classical Young symmetrizers (see, e.g., [33]); the theory of tensor spaces under the action of general linear groups and symmetric groups, in which standard symmetrized bitableaux give bases of the symmetry classes (see, e.g., [40]); the theory of spaces of polynomial functions of several vector variables under the action of general linear groups, in which the basis of standard symmetrized bitableaux becomes the Gordan–Capelli series (in the sense of [39]), and Young–Capelli symmetrizers lead to an explicit form of the Capelli-Deruyts expansion (in the sense of [32]); the Berele–Regev theory of super tensor spaces under the action of general linear Lie superalgebras and symmetric groups [1, 2].

For a detailed description of the relations of the theory with classical invariant theory, as well as of the relations with the representation theory of general linear Lie superalgebras, we refer to [6].

Let 𝕂 a be field of characteristic zero, L and P two finite ℤ_{2}-graded sets, pl(L) and pl(P) the corresponding general linear Lie superalgebras, Super[L|P] the supersymmetric algebra over L × P, and Super_{n}[L|P] the ℤ−homogeneous n−th component of this algebra. In [8], Brini, Palareti and Teolis defined symmetrized bitableaux in the space Super_{n}[L|P], proved that the superstandard symmetrized bitableaux form a linear basis of this space, and got from this basis an explicit complete decomposition of Super_{n}[L|P] as a semisimple pl(L)−module. Roughly speaking, the proof of the spanning property relies on the Grosshans–Rota–Stein Superstraightening Law [26] and the proof of the linear independence relies on the Berele–Regev theory [1, 2]. In [9], Brini and Teolis constructed a complete decomposition of the operator algebra induced by the action of pl(L) as a semisimple algebra, by defining the Young–Capelli symmetrizers and proving a Triangularity Theorem for the action of the superstandard Young–Capelli symmetrizers on the superstandard symmetrized bitableaux. The proof of the Triangularity Theorem relies on the Berele–Regev theory [1, 2] and on the properties of Gale–Ryser interpolating matrices [26].

In the last years, the theory has been given a new architecture. In [12], it is outlined how the complete decompositions of the semisimple pl(L)-module Super_{n}[L|P] and of the semisimple operator algebra induced by the pl(L)-action can be simultaneously derived by the Triangularity Theorem for the action of the superstandard Young– Capelli symmetrizers on the superstandard symmetrized bitableaux. In [13] it was shown that the Triangularity Theorem can be directly derived from a Triangularity Lemma and a Nondegeneracy Lemma, and it was announced that these lemmas have elementary combinatorial proofs that will appear elsewhere. We remark that, while the Triangularity Lemma is a refinement of a result in [9] which in [9] had already an elementary combinatorial proof, the Nondegeneracy Lemma, implicitly stated in [8, 9], is exactly the point in which the theory was dependent on the Berele–Regev theory.

In [6], it is given a strong form of the Nondegeneracy Lemma, together with a rather indirect proof based on the ordinary representation theory of the symmetric group.

In this paper, we give a detailed exposition of the foundations of the theory, as well as an outline of the main results. To wit: we give, through Capelli’s method of virtual variables, the definitions of the basic objects of the theory; we establish three new propositions on virtual expressions; we give the statements of the Triangularity Lemma, of the Nondegeneracy Lemma and of the Superstraightening Law; we give an outline of the theory till the theorems on complete decompositions; we prove from scratch the three propositions and the two lemmas. All these proofs are new; the proof of the Triangularity Lemma is based on the easy part of the Gale–Ryser Theorem; the proof of the weak form of the Nondegeneracy Lemma is obtained by a direct combinatorial argument; the proof of the strong form of the Nondegeneracy Lemma is based on the Superstraightening Law. With these proofs, we complete the process of giving the theory an elementary combinatorial foundation.

1. Words and Tableaux

In this section we recall some notions pertaining to words and tableaux on a ℤ_{2}- graded set. Here, our aim is just to fix terms and notations that will be used throughout.

Let A be a set endowed with a disjoint union decomposition A = A^{+} ⊔ A^{−} into a subset A^{+} of positive elements and a subset A^{−} of negative elements; such a decomposition may be encoded by the mapping | | : A → ℤ_{2} with values in the two-element field ℤ_{2} which sends the positive and negative elements to 0 and 1, respectively. A set with this structure is called a signed set, or a ℤ_{2}-graded set.

Let w = x_{1}x_{2} ⋯ x_{n} be a word of length l(w) = n on A. The support of w is the set sp(w) of the elements of A that occur in w; the content of w is the family c_{w} of the occurrences c_{w}(a) of the elements a of A in w.

Let w = x_{1}x_{2} ⋯ x_{n} and w′ = x′_{1}x′_{2} ⋯ x′_{n} be two words of the same length on A. We say that w and w′ are equivalent, and we write

w ≡ w′,

if w and w′ have the same content. This happens if and only if there is a permutation of the index set {1, …, n} which carries w to w′. If there is a ℤ_{2}−grade preserving bijection A → A which carries w to w′, we say that w and w′ are isomorphic, and we write

w ≃ w′.

We say that two sequences (w, z, …) and (w′, z′, …) of words are isomorphic, and we write (w, z, …) ≃ (w′, z′, …), if there is a ℤ_{2}−grade preserving bijection A → A which carries w to w′, z to z′, ….

A partition is a weakly decreasing sequence λ = (λ_{1}, λ_{2}, …, λ_{p}) of nonnegative integers; the nonnegative integer ∑ λ_{i} is called the weight of the partition, and is denoted by |λ|; the sentence “λ has weight n” is rephrased also as “λ is a partition of n”, and written λ ⊢ n. A partition λ = (λ_{1}, λ_{2}, …) is dominated by a partition μ = (μ_{1}, μ_{2}, …) if

λ_{1} + ⋯ + λ_{i} ≤ μ_{1} + ⋯ + μ_{i}, for all i = 1, 2, ….

The relation of dominance is a partial order on partitions, and will be denoted by the symbol ≤ . We take for granted some basic notions pertaining to partitions, such as the notion of skew partition, horizontal strip, and vertical strip (see, e.g., [31, pp. 1–5]).

A tableau over the signed set A is a sequence

S = (w_{1}, w_{2}, …, w_{p})

of words w_{i} = w_{i1}w_{i2} ⋯ w_{iλ_{i}} of weakly decreasing lengths. The partition λ = (λ_{1}, λ_{2}, …, λ_{p}) is called the shape of S, and is denoted by sh(S). The word w obtained by juxtaposition of the words w_{i},

w = w_{1}w_{2} ⋯ w_{p} = w(S)

is called the row word of S. If n is the length of w, then λ is a partition of n, and we say that the tableau S has size n.

The tableau S can be represented by writing the words w_{1}, w_{2}, …, w_{p} with their first entries directly under one another

S = w_{11}w_{12} . . . . . . . . . w_{1λ_{1}} ; w_{21}w_{22} . . . . . . w_{2λ_{2}} ; ⋮ ; w_{p1}w_{p2} … w_{pλ_{p}}

or, in matrix-like notation, S = (w_{ij})_{{i=1,...,p \atop j=1,...,λ_{i}}}.

The content of the tableau S is the family c_{S} of the occurrences c_{S}(a) of the elements a of A in S; the row-content of S is the family c_{S}^{r} of the occurrences c_{S}^{r}(a, i) of the elements a of A in the rows w_{i} of S; the column-content of S is the family c_{S}^{c} of the occurrences c_{S}^{c}(a, j) of the elements a of A in the columns z_{j} of S.

If two tableaux S and S′ have the same content, then we say that they are equivalent, and we write S ≡ S′; if S and S′ have the same row-content, we say that they are rowequivalent, and we write

S ≡_{r} S′;

if S and S′ have the same column-content, we say that they are column-equivalent, and we write S ≡_{c} S′.

We denote the sets of all the tableaux over A of size n, of shape λ, respectively by the symbols

𝔱_{n}[A], 𝔱_{λ}[A];

when no confusion arises, we will suppress the specification [A].

We assume now that the signed set A is a linearly ordered set, of the same order type of a subset of the natural numbers. A tableau S over A is called standard if

• each row of S is weakly increasing, with no repetition of negative symbols,

• each column of S is weakly increasing, with no repetition of positive symbols.

It is known that, if |A^{+}| = p and |A^{−}| = q, then there is a standard tableau of shape λ with entries in A if and only if λ_{p+1} < q + 1. The set of these partitions is called the hook set of the signed set A, and is denoted by H(A).

Let S be a standard tableau over A. For every positive integer p, the object S^{(p)} obtained by considering in S only the first p symbols of A is a standard tableau; the shapes of these tableaux form a sequence

sh(S^{(1)}) ⊆ sh(S^{(2)}) ⊆ … = sh(S) = sh(S) = …,

such that the skew partition sh(S^{(p)})/sh(S^{(p−1)}) is an horizontal strip or a vertical strip according to the p−th symbol of A is positive or negative. Any sequence of partitions of this type is achieved by exactly one standard tableau.

We use the partial order on the set of all standard tableaux over A defined by setting

S ≤ T if and only if S ≡ T, and sh(S^{(p)}) ≤ sh(T^{(p)}), for all p = 1, 2, …

This partial order on standard tableaux lies between the usual linear order on tableaux [26] and the hyperdominance order on tableaux [20].

We denote the sets of all the standard tableaux over A of size n, of shape λ, respectively, by the symbols

𝔰𝔱_{n}[A], 𝔰𝔱_{λ}[A].

When no confusion arises, we will suppress the specification [A].

2. Setting

In this section, we introduce the supersymmetric algebra Super[L|P] on the direct product of two finite ℤ_{2}-graded sets L, P, left and right superpolarization operators on this algebra, and the bimodules we are interested in. We recall all what we need; we refer to [34, Chapter 0], for the basic superalgebraic notions, and to [6, Sections 3 and 4], for further aspects, namely the representation-theoretical ones.

2.1. The Superalgebra Super[A]. Let 𝕂 be a field of characteristic 0; by vector space, algebra, linear mapping, etc., we will always mean 𝕂−vector space, 𝕂−algebra, 𝕂−linear mapping, etc..

A ℤ_{2}-grading on a vector space V is a decomposition V = V_{0}⊕V_{1} of V as a direct sum of two subspaces V_{i}, with i ∈ ℤ_{2}. The elements of V_{0} and V_{1} are said to be homogeneous; the grade of an homogeneous element v ∈ V will be denoted by |v|. A ℤ_{2}-grading on an algebra A is a decomposition A = A_{0} ⊕ A_{1} of A as a direct sum of two subspaces A_{i}, with i ∈ ℤ_{2}, such that A_{i} A_{j} ⊆ A_{i+j}, for all i, j ∈ ℤ_{2}. The superalgebra A is called ℤ_{2}-graded commutative whenever

ab = (−1)^{|a||b|}ba,

for every homogeneous elements a, b in A.

A linear mapping f : V → W from a ℤ_{2}-graded vector space V = V_{0} ⊕ V_{1} from a ℤ_{2}-graded vector space W = W_{0} ⊕ W_{1} is said to be homogeneous of degree i ∈ ℤ_{2} when f[V_{j}] ⊂ W_{j+i}, for all j ∈ ℤ_{2}. Given a ℤ_{2}-graded algebra A, a linear mapping D : A → A, homogeneous of degree |D|, is said to be a left superderivation whenever

D(ab) = D(a)b + (−1)^{|D||a|}aD(b),

for every homogeneous elements a, b in A; a linear mapping A ← A : D, homogeneous of degree |D|, is said to be a right superderivation whenever

(ab)D = (−1)^{|b||D|}(a)Db + a(b)D,

for every homogeneous elements a, b in A.

Let A be a ℤ_{2}-graded set. The free monoid Mon[A] on a A has a natural ℤ_{2}-grading given by |w| = |x_{1}| + |x_{2}| + ⋯ + |x_{n}|, for every word w = x_{1}x_{2} ⋯ x_{n} on A. The supersymmetric algebra Super[A] generated by the the ℤ_{2}-graded set A is the quotient algebra

Super[A] = \frac{𝕂[Mon[A]]}{I[A]}

of the semigroup algebra 𝕂[Mon[A]] of the free monoid Mon[A] with respect to the two-sided ideal I[A] generated by the elements of the form

xy − (−1)^{|x||y|}yx, x, y ∈ A.

A word w in Mon[A] is nonzero in Super[A] if and only if each negative symbol occurs in w at most once; two such words give rise, up to a sign, to the same element of Super[A] if and only if they have the same content, i.e., they have are equivalent. By choosing one word for each admissible content, one gets a linear basis of this algebra. We define the sign ϵ_{w,w′} ∈ {−1, 1} of two nonzero equivalent words w, w′, by setting

w = ϵ_{w,w′} w′.

The algebra Super[A] has a natural ℤ_{2}-grading, given by

Super[A] = ⟨w ∈ Mon[A]; |w| = 0⟩ ⊕ ⟨w ∈ Mon[A]; |w| = 1⟩
= Super[A]_{0} ⊕ Super[A]_{1},

and it is ℤ_{2}-graded commutative. This algebra has also a natural ℤ-grading, given by

Super[A] = ⊕_{n∈ℕ}⟨w ∈ Mon[A]; l(w) = n⟩
= ⊕_{n∈ℕ}Super_{n}[A],

and the two gradings are consistent.

For every x, y in A, there is exactly one left superderivation,

D_{xy} : Super[A] → Super[A],

homogeneous of degree |xy| = |x|+|y|, called the left superpolarization of y to x, which takes the values D_{xy}(u) = δ_{yu} x on the generators of Super[A]. For every x, y in A, there is exactly one right superderivation

Super[A] ← Super[A] : _{xy}D,

homogeneous of degree |xy| = |x| + |y|, called the right superpolarization of x to y, which takes the values (u) _{xy}D = δ_{ux} y on the generators of Super[A].

2.2. The Superalgebra Super[L|P]. Let L, P be two finite ℤ_{2}-graded sets, endowed with a linear order. We consider the copy (L|P) = {(x|y); x ∈ L, y ∈ P} of the Cartesian product L × P, where the ℤ_{2}−grade of an ordered pair is defined as the sum of the grades of its components: |(x|y)| = |xy| = |x| + |y|. Notice that (x|y) is positive or negative if x and y have equal or different degrees, respectively.

Our discussion will take place in the supersymmetric algebra Super[L|P] generated by the ℤ_{2}-graded set (L|P), that is, the quotient algebra

Super[L|P] = \frac{𝕂[Mon[L|P]]}{I[L|P]}

of the semigroup algebra 𝕂[Mon[L|P]] of the free monoid Mon[L|P] with respect to the two-sided ideal I[L|P] generated by the elements of the form

(x|y)(x′|y′) − (−1)^{|xy||x′y′|}(x′|y′)(x|y), (x|y), (x′|y′) ∈ (L|P).

In particular, this algebra is linearly generated by the “monomials”

(x_{1}|y_{1})(x_{2}|y_{2}) ⋯ (x_{n}|y_{n})

in the double symbols (x_{i}|y_{i}) in (L|P), and a monomial is nonzero if and only if it is square-free in the negative double symbols. The algebra Super[L|P] has a natural ℤ_{2}-grading

Super[L|P] = Super[L|P]_{0} ⊕ Super[L|P]_{1},

with respect to which it is ℤ_{2}-graded commutative, and has also a natural ℤ-grading

Super[L|P] = ⊕_{n∈ℕ}Super_{n}[L|P],

consistent with the former grading.

For every x, y in L, there is exactly one left superderivation,

D_{xy} : Super[L|P] → Super[L|P],

homogeneous of degree |xy| = |x|+|y|, called the left superpolarization of y to x, which takes the values D_{xy}(u|v) = δ_{yu} (x|v) on the generators of Super[L|P]. In particular, the action of a left superpolarization on a monomial is given by

D_{xy} ((x_{1}|y_{1})(x_{2}|y_{2}) ⋯ (x_{n}|y_{n}))

= ∑ ϵ_{i} δ_{yx_{i}} (x_{1}|y_{1}) ⋯ (x_{i−1}|y_{i−1})(x|y_{i})(x_{i+1}|y_{i+1}) ⋯ (x_{n}|y_{n}),

where ϵ_{i} = (−1)^{|xy||x_{1}y_{1}···x_{i−1}y_{i−1}|}. Left superpolarizations satisfy the Lie superbracket relations

D_{xy}D_{zt} − (−1)^{|xy||zt|}D_{zt}D_{xy} = δ_{yz} D_{xt} − (−1)^{|xy||zt|}δ_{tx} D_{zy}.

For every x, y in P, there is exactly one right superderivation

Super[L|P] ← Super[L|P] : _{xy}D,

homogeneous of degree |xy| = |x|+|y|, called the right superpolarization of x to y, which takes the values (u|v) _{xy}D = δ_{vx} (u|y) on the generators of Super[L|P]. In particular, the action of a right superpolarization on a monomial is given by

((x_{1}|y_{1})(x_{2}|y_{2}) ⋯ (x_{n}|y_{n})) _{xy}D

= ∑ η_{i} δ_{y_{i}x} (x_{1}|y_{1}) ⋯ (x_{i−1}|y_{i−1})(x_{i}|y)(x_{i+1}|y_{i+1}) ⋯ (x_{n}|y_{n}), where η_{i} = (−1)^{|x_{i+1}y_{i+1}···x_{n}y_{n}||xy|}. Right superpolarizations satisfy the Lie superbracket relations

xyD _{zt}D − (−1)^{|xy||zt|} _{zt}D _{xy}D = δ_{yz} _{xt}D − (−1)^{|xy||zt|} δ_{tx} _{zy}D.

Left superpolarization an right superpolarizations commute:

(D_{xy} \underline{ξ}) _{zt}D = D_{xy} (\underline{ξ} _{zt}D) , \underline{ξ} ∈ Super[L|P].

Henceforth, n will be a fixed positive integer. The finite-dimensional space Super_{n}[L|P] is invariant with respect to any left polarization D_{xy} with x, y ∈ L, as well as with respect to any right polarization_{zt}D with z, t ∈ P; the restrictions of these polarizations to the space Super_{n}[L|P] generate two subalgebras B_{n}[L|P], [L|P]_{n}B. of the algebra of linear endomorphisms of Super_{n}[L|P].

We are interested in the bimodule

B_{n}[L|P] · Super_{n}[L|P] · [L|P]_{n}B.

3. Basic Objects

The basic objects in the bimodule B_{n}[L|P] · Super_{n}[L|P] · [L|P]_{n}B are defined as the result of expressions containing certain ℤ_{2}-graded “virtual variables”. The use of ℤ_{2}-graded virtual variables allows us to provide simple and transparent definitions of objects with desired symmetry properties; the values of the virtual variables are irrelevant. This leads to work in a wider bimodule and to define basic objects in this wider bimodule.

The formal setting is the following. The finite ℤ_{2}-graded sets L and P are identified with two disjoint subsets of a ℤ_{2}-graded set X consisting of countably many positive and countably many negative elements. We assume that the linear orders defined on the sets L, P are extended to a linear order on the set X , which makes X order-isomorphic to the set ℕ of natural numbers. Then, we consider the supersymmetric algebra Super[X |X] on the ℤ_{2}-graded set X × X , left superpolarizations D_{xy} and right superpolarizations _{xy}D, with x, y ∈ X . These polarizations leave invariant the ℕ−homogeneous component Super_{n}[X |X], the restrictions of these polarizations to this space generate two subalgebras B_{n}[X] and [X]_{n}B of the algebra of linear endomorphisms of Super_{n}[X |X], and we consider the bimodule

B_{n}[X] · Super_{n}[X |X] · [X]_{n}B.

For any linear endomorphism φ of the algebra Super[X |X] that stabilizes the subspace Super_{n}[X |X], the linear endomorphism of the space Super_{n}[X |X] induced by φ will be denoted by φ|_{n} . In this bimodule the expressions which define the basic objects of the theory are given. The consistency of these definitions will follow from the general propositions in the next section.

For any x, y ∈ X , we will use the following short notations

\underrightarrow{xy}, \underline{xy}, \underleftarrow{xy}, for D_{xy}, (x|y), _{xy}D;

when no confusion may arise, we will write simply xy for \underrightarrow{xy} or \underleftarrow{xy}.

Let P = p_{1} ⋯ p_{n}, Q = q_{1} ⋯ q_{n}, be a pair of words of length n on X .

The left Capelli row H[P |Q], the biproduct (P |Q), and the right Capelli row [P |Q]H, are defined as follows:

H[P |Q] = \underrightarrow{p_{1}a} ⋯ \underrightarrow{p_{n}a} \underrightarrow{aq_{1}} ⋯ \underrightarrow{aq_{n}}|_{n} ,
(P |Q) = p_{1}a ⋯ p_{n}a (\underline{aq_{1}} ⋯ \underline{aq_{n}}) = (\underline{p_{1}a} ⋯ \underline{p_{n}a}) aq_{1} ⋯ aq_{n},
[P |Q]H = \underleftarrow{p_{1}a} ⋯ \underleftarrow{p_{n}a} \underleftarrow{aq_{1}} ⋯ \underleftarrow{aq_{n}}|_{n} ,

where a ∈ X is a positive symbol occurring neither in P nor in Q (see [6, Subsections 6.3, 7.2]). Notice that, under transposition of two adjacent entries r and s in P or in Q, each of these objects is transformed into itself, up to the sign (−1)^{|r||s|}. Biproducts are far-reaching generalizations of determinants and permanents; they were first defined, as objects satisfying all possible Laplace expansions, in ([26, Chapter 2]). In ([8]), the definition of biproducts in terms of virtual variables was given. Capelli rows are far-reaching generalizations of the classical Capelli operators (see [40, Chapter 2, Section 4]); they were defined, in virtual form, in [9]. The classical Capelli operators were defined in this way in [28].

The left Capelli ∗−row H^{∗}[P |Q], the ∗−biproduct (P |Q)^{∗}, and the right Capelli ∗−row [P |Q]^{∗}H, are defined in an analogous way, the difference being that now a is a negative symbol.

To any pair (P, Q) of words P = p_{1} ⋯ p_{m}, Q = q_{1} ⋯ q_{m}, of the same length on X we associate a left polarization monomial, a monomial, and a right polarization monomial

\underrightarrow{P Q} = ∏_{i=1}^{m} \underrightarrow{p_{i}q_{i}},  \underline{P Q} = ∏_{i=1}^{m} \underline{p_{i}q_{i}},  \underleftarrow{P Q} = ∏_{i=1}^{m} \underleftarrow{p_{i}q_{i}};

when no confusion may arise, we will write simply simply P Q for \underrightarrow{P Q} or \underleftarrow{P Q}. To any pair of tableaux of the same shape on X , we associate their row-words, and then the corresponding monomials. Specifically, to any pair (P, Q) of tableaux

P = p_{11} · · · · · · · · · p_{1λ_{1}} ; p_{21} · · · · · · p_{2λ_{2}} ; ⋮ ; p_{r1} ⋯ p_{rλ_{r}} , Q = q_{11} · · · · · · · · · q_{1λ_{1}} ; q_{21} · · · · · · q_{2λ_{2}} ; ⋮ ; q_{r1} ⋯ q_{rλ_{r}}

of the same shape λ on X we associate a left polarization monomial, a monomial and a right polarization monomial

\underrightarrow{P Q} = ∏_{f=1}^{r} ∏_{g=1}^{λ_{f}} \underrightarrow{p_{fg}q_{fg}},  \underline{P Q} = ∏_{f=1}^{r} ∏_{g=1}^{λ_{f}} \underline{p_{fg}q_{fg}},  \underleftarrow{P Q} = ∏_{f=1}^{r} ∏_{g=1}^{λ_{f}} \underleftarrow{p_{fg}q_{fg}};

when no confusion may arise, we will write simply P Q for \underrightarrow{P Q} or \underleftarrow{P Q}.

In the following, a crucial role will be played by two types of tableaux, which are are used to induce (super)symmetry along rows and (super)skew-symmetry along columns. These tableaux are:

• tableaux filled with positive symbols, such that any two symbols in the same row are equal and any two symbols in the same column are distinct:

C = a_{1}a_{1} · · · · · · · · · a_{1} ; a_{2}a_{2} · · · · · · a_{2} ; ⋮ ; a_{r}a_{r} ⋯ a_{r}  |a_{i}| = 0,  a_{i} ≠ a_{j} for i ≠ j.

These tableaux are said to be of coDeruyts type, or coDeruyts for short. The symbol C and its variations like C′, Ċ, C′′, … will always denote tableaux of coDeruyts type.

• tableaux filled with negative symbols, such that any two symbols in the same column are equal and any two symbols in the same row are distinct:

D = b_{1}b_{2} · · · · · · · · · b_{λ_{1}} ; b_{1}b_{2} · · · · · · b_{λ_{2}} ; ⋮ ; b_{1}b_{2} ⋯ b_{λ_{r}}  |b_{i}| = 1,  b_{i} ≠ b_{j} for i ≠ j.

These tableaux are said to be of Deruyts type, or Deruyts for short. The symbol D and its variations like D′, Ḋ, D′′, … will always denote tableaux of Deruyts type.

Let (P, Q) be a pair of tableaux of the same shape λ ⊢ n on X .

The left Capelli bitableau H[P |Q], the bitableau (P |Q), and the right Capelli bitableau [P |Q]H are defined as follows:

H[P |Q] = \underrightarrow{P C} \underrightarrow{CQ} |_{n} ,
(P |Q) = P C \underline{CQ} = \underline{P C} CQ,
[P |Q]H = \underleftarrow{P C} \underleftarrow{CQ} |_{n} ,

where C is any coDeruyts tableau of shape λ whose support is disjoint from those of P and Q (see [6, Subsections 6.3, 7.4]). Under transposition of two adjacent entries r and s in a row of P or in a row of Q, each of these objects is transformed into itself, up to the sign (−1)^{|r||s|}. If P and Q are columns, then these objects are, up to a sign, the left polarization monomial, the monomial, and the right polarization monomial associated to the pair of words (P, Q); if P and Q are rows, then these objects are the left Capelli row, the biproduct, and the right Capelli row associated to the pair of words (P, Q). Bitableaux were first defined, as products of biproducts , in [26]. In [8], the definition of bitableaux in terms of virtual variables was given. Capelli bitableaux were defined, in virtual form, in [9].

The left Young–Capelli symmetrizer γ(P, Q ), the symmetrized bitableau (P | Q ), and the right Young–Capelli symmetrizer (P, Q )γ, are defined as follows:

γ(P, Q ) = \underrightarrow{P C} \underrightarrow{CD} \underrightarrow{DQ} |_{n} ,
(P | Q ) = P C CD \underline{DQ} = P C \underline{CD} DQ = \underline{P C} CD DQ,
(P, Q )γ = \underleftarrow{P C} \underleftarrow{CD} \underleftarrow{DQ} |_{n} ,

where C and D are a coDeruyts and a Deruyts tableau of shape λ, C has support disjoint from that of P, D has support disjoint from that of Q (see [6, Subsections 9.1, 10.1]). Notice that, under transposition of two entries in a row of P or in a column of Q, each of these objects is transformed into itself, up to a sign. Symmetrized bitableaux were first defined, in virtual form, in [8]. They are far-reaching generalizations of the members of the classical Gordan–Capelli series (see [39], and [18, 19] for ramifications in other directions). Young–Capelli symmetrizers were defined, in virtual form, in [9]. They are far-reaching generalizations of the classical Young symmetrizers.

The left Capelli ∗−bitableau H^{∗}[P |Q], the ∗−bitableau (P |Q)^{∗}, and the right Capelli ∗−bitableau [P |Q]^{∗}H are defined by

H^{∗}[P |Q] = \underrightarrow{P D} \underrightarrow{DQ} |_{n} ,
(P |Q)^{∗} = P D \underline{DQ} = \underline{P D} DQ,
[P |Q]^{∗}H = \underleftarrow{P D} \underleftarrow{DQ} |_{n} ,

where D is any Deruyts tableau of shape λ whose support is disjoint from those of P and Q [8, 9].

4. Virtual Expressions

The left polarization monomials, the monomials and the right polarization monomials entering into the definitions of the basic objects in the bimodule B_{n}[X]·Super_{n}[X |X]· [X]_{n}B are always associated to pairs of words of length n over X with disjoint supports. This leads to consider the class of all the expressions obtained by combining these monomials and polarization monomials. For the sake of brevity, we refer to these expressions as virtual expressions of step n.

Any virtual expression of step n

P Q \underline{RS}

yields an element of Super_{n}[X |X], which can be written in the form

P Q \underline{RS} = ∑_{P′} ξ_{P′} \underline{P′S},

where P′ ranges among the words equivalent to P and the coefficients ξ_{P′} are integers. This element depends on the central pair (Q, R) only through its isomorphism type, and is zero unless Q and R have the same content. Analogous remarks can be made for any virtual expression \underline{P Q} RS.

Proposition 1. Virtual expressions of step n satisfy the identity

P Q \underline{RS} = \underline{P Q} RS.

This proposition is new. We give a proof in Subsection 9.2.

More generally, the u virtual expressions of step n

P_{1}Q_{1} ⋯ P_{i−1}Q_{i−1} \underline{P_{i}Q_{i}} P_{i+1}Q_{i+1} ⋯ P_{u}Q_{u}, i = 1, …, u

yield the same element in Super_{n}[X |X], which can be written as

∑_{P′,Q′} ζ_{P′Q′} \underline{P′Q′},

where P′ ranges among the words equivalent to P_{1}, Q′ ranges among the words equivalent to Q_{u}, and the coefficients ζ_{P′Q′} are integers. This element depends on the central pairs (Q_{1}, P_{2}), …, (Q_{u−1}, P_{u}) only trough their isomorphism type, and is zero unless in each of these pairs the words have the same content. This property ensures that all the basic objects for Super_{n}[X |X] are well-defined, since the row-words of any two coDeruyts tableaux of the same shape are isomorphic, as well as the row-words of any two Deruyts tableaux of the same shape are isomorphic.

Starting from sequences of pairs of words of length n on X with disjoint supports, one can obtain expressions with values in Super_{n}[X |X], as well as expressions with values in B_{n}[X] or in [X]_{n}B. It turns out that the three types of expressions associated to the same set of sequences satisfy the same linear relations.

Proposition 2. Let (P_{1}^{i}, Q_{1}^{i}), (P_{2}^{i}, Q_{2}^{i}), …, (P_{m_{i}}^{i}, Q_{m_{i}}^{i}), for i = 1, …, u be a series of sequences of pairs of words of length n on X , such that any two words in the same pair have disjoint supports, let ϑ_{i} be scalars, and let j_{i} be indexes with 1 ≤ j_{i} ≤ m_{i}. Then the following equalities are equivalent

∑_{i} ϑ_{i} \underrightarrow{P_{1}^{i}Q_{1}^{i}} \underrightarrow{P_{2}^{i}Q_{2}^{i}} ⋯ \underrightarrow{P_{m_{i}}^{i}Q_{m_{i}}^{i}} |_{n} = \underrightarrow{0},

∑_{i} ϑ_{i} P_{1}^{i}Q_{1}^{i} ⋯ P_{j_{i}−1}^{i}Q_{j_{i}−1}^{i} \underline{P_{j_{i}}^{i}Q_{j_{i}}^{i}} P_{j_{i}+1}^{i}Q_{j_{i}+1}^{i} ⋯ P_{m_{i}}^{i}Q_{m_{i}}^{i} = \underline{0},

∑_{i} ϑ_{i} \underleftarrow{P_{1}^{i}Q_{1}^{i}} \underleftarrow{P_{2}^{i}Q_{2}^{i}} ⋯ \underleftarrow{P_{m_{i}}^{i}Q_{m_{i}}^{i}} |_{n} = \underleftarrow{0}.

This proposition is new. We give a proof in Subsection 9.3.

By interchanging left and right, one gets from each of the previous equalities an equivalent equality; for example, from the second equality one gets the equivalent equality

∑_{i} ϑ_{i} \overline{Q}_{m_{i}}^{i}\overline{P} _{m_{i}}^{i} ⋯ \overline{Q}_{j_{i}+1}^{i}\overline{P} _{j_{i}+1}^{i} \underline{\overline{Q}_{j_{i}}^{i}\overline{P} _{j_{i}}^{i}} \overline{Q}_{j_{i}−1}^{i}\overline{P} _{j_{i}−1}^{i} ⋯ \overline{Q}_{1}^{i}\overline{P} _{1}^{i} = \underline{0},

where, for each word R = r_{1}r_{2} ⋯ r_{n}, we have set \overline{R} = r_{n} ⋯ r_{2}r_{1}.

From Proposition 2 it follows in particular that the value in B_{n}[X] of any virtual expression of step n

\underrightarrow{P_{1}Q_{1}} \underrightarrow{P_{2}Q_{2}} ⋯ \underrightarrow{P_{u}Q_{u}} |_{n}

depends on the pairs (Q_{1}, P_{2}), …, (Q_{u−1}, P_{u}) only trough their isomorphism type, and is zero unless in each of these pairs the words have the same content. This ensures that all the basic objects for B_{n}[X] are well-defined. Analogous remarks can be made for expressions with values in [X]_{n}B.

The expansions of virtual expressions with values in B_{n}[X] can be given in the following way.

Proposition 3. Any virtual expression \underrightarrow{P_{1}Q_{1}} \underrightarrow{P_{2}Q_{2}} ⋯ \underrightarrow{P_{u}Q_{u}} |_{n} of step n can be written as

\underrightarrow{P_{1}Q_{1}} \underrightarrow{P_{2}Q_{2}} ⋯ \underrightarrow{P_{u}Q_{u}} |_{n} = ∑_{P′,Q′} η_{P′,Q′} \underrightarrow{P′Q′}|_{n} ,

where P′, Q′ range among the pairs of words of the same length equivalent to subwords of P_{1}, Q_{u}, and the coefficients η_{P′,Q′} are integers.

This proposition is new (compare with [6, Theorems 6.1, 6.2, 6.3]). An analogous result holds for virtual expressions with values in [X]_{n}B. We give a proof in Subsection 9.4.

5. Triangularity, Nondegeneracy, and Straightening

5.1. Triangularity and Nondegeneracy Lemmas. Our starting point is the following instance of the “easy part” of the Gale–Ryser Theorem.

Scholion 1. Let (A, B) be a pair of words of the same length on X . If A has the same content of a coDeruyts tableau C, and B has the same content of a Deruyts tableau D, then

\underline{AB} = \{ \begin{matrix}\underline{0} & if sh(C) ≰ sh(D) \\ η \underline{CD} & if sh(C) = sh(D) \end{matrix} ,

with η ∈ {0, ±1}.

For the sake of completeness, we give a proof in Subsection 10.1.

Lemma 1 (Triangularity Lemma I; see [6, Lemma 8.1]). Let (A_{1}, B_{1}), (A_{2}, B_{2}), …, (A_{s}, B_{s}) be pairs of words of the same length on X , where any two words in the same pair have disjoint supports, and let 1 ≤ i ≤ s. If A_{1} has the same content of a coDeruyts tableau C and B_{s} has the same content of a Deruyts tableau D, then

A_{1}B_{1} ⋯ A_{i−1}B_{i−1} \underline{A_{i}B_{i}} A_{i+1}B_{i+1} ⋯ A_{s}B_{s} = \{ \begin{matrix}\underline{0} & if sh(C) ≰ sh(D) \\ ϑ \underline{CD} & if sh(C) = sh(D) \end{matrix} ,

where ϑ is an integer.

Lemma 2 (Triangularity Lemma II; see [6, Lemma 8.2]). Let (C, S), (T, D) be pairs of tableaux of the same size, where tableaux in the same pair have the same shape and disjoint supports, C is coDeruyts, S and T are standard, and D is Deruyts. Then

CS \underline{T D} = \{ \begin{matrix}\underline{0} & if S ≰ T \\ ϑ \underline{CD} & if S = T \end{matrix} ,

where ϑ = ±c_{S}^{r}!c_{S}^{c}! is the product of the factorial of the row-content and the factorial of the column-content of P.

These lemmas are refinements of Propositions 1 and 2 in [9]. They can be regarded as virtual forms of the propositions on Capelli operators in the sense of [22, Theorem 2.1], (see also [21, 20, 27]). In [6], elementary combinatorial proofs of these lemmas are announced. We fulfill this announcement by providing proofs in Subsections 10.2, 10.4.

Let P, Q be two tableaux of the same shape λ ⊢ n. From Lemma 1 it follows that:

• there is exactly one integer ϑ_{P Q}^{+−} such that

CP \underline{QD} = ϑ_{P Q}^{+−} \underline{CD},

for every coDeruyts tableau C of shape λ, with support disjoint from that of P, and for every Deruyts tableau D of shape λ, with support disjoint from that of Q. This integer is called the (+−) symmetry coefficient of the two tableaux P and Q;

• for any φ ∈ B_{n}[X], there is exactly one integer ϑ_{P Q}^{+−}(φ) such that

CP (φ \underline{QD}) = ϑ_{P Q}^{+−}(φ) \underline{CD},

for every coDeruyts tableau C of shape λ, with support disjoint from that of P, and for every Deruyts tableau D of shape λ, with support disjoint from that of Q.

Example 1. Given two multilinear tableaux P and Q of the same shape, filled with positive symbols of X , we have: if there are two symbols occurring in the same row of P and in the same column of Q, then ϑ_{P Q}^{+−} = 0; otherwise, there is exactly one multilinear tableau W, of the same shape of P and Q, such that P ≡_{r} W ≡_{c} Q, and ϑ_{P Q}^{+−} = ϵ_{Q,W} , where ϵ_{Q,W} is the sign of the permutation that transforms Q into W.

Given a partition λ ⊢ n, we have the infinite matrices

Θ_{t_{λ}}^{+−} = [ϑ_{P Q}^{+−}]_{P,Q∈t_{λ}} ,

whose entries are the symmetry coefficients of pairs of tableaux of shape λ over X . Since ϑ_{P Q}^{+−} is zero unless P and Q have the same content, we have that this matrix is a block diagonal matrix, with diagonal blocks parametrized by all the possible multisets of order n over X; furthermore, if two multisets have the same statistic of multiplicities, then the corresponding blocks are equal, up to permutation of rows and columns.

As a consequence of Lemma 2, for every partition λ ⊢ n the (+−)symmetry coefficients of the standard tableaux satisfy the nondegenerate triangularity relations

ϑ_{P Q}^{+−} \{ \begin{matrix}= 0 & if P ≰ Q \\ ≠ 0 & if P = Q \end{matrix} .

Therefore, the matrix Θ_{st_{λ}}^{+−} = [ϑ_{P Q}^{+−}]_{P,Q,∈st_{λ}} is nonsingular. The inverse matrix is still a block diagonal matrix with the same structure. Its entries are called the Rutherford coefficients and are denoted by ϱ_{ST}^{+−}. In the case of Example 1, the Rutherford coefficients are the classical ones [33].

Lemma 3 (Nondegeneracy Lemma; see [6, Lemma 8.3]). Let C, Ḋ, Ċ, D be coDeruyts and Deruyts tableaux of the same shape λ ⊢ n. Then

CḊ \underline{ḊĊ} ĊD = k_{λ} \underline{CD},

where k_{λ} = (−1)^{\binom{n}{2}} h_{λ}, and h_{λ} is the product of the hook lengths of λ.

In [6], this lemma is derived, in a rather indirect way, by the theorem of essential idempotency of the classical Young Symmetrizers (see, e.g., [4, Chapter IV, Theorem 3.1]). In Subsections 11.3, 11.4, we give a proof based on the Superstraightening Law. We remark that the significance of this lemma for the general theory relies in the statement that k_{λ} ≠ 0. We give a proof of this weaker statement, based on a direct combinatorial argument, in Subsection 11.2.

We remark that all the statements in this subsection yield relations between virtual expressions. Thus each of these statements yields other equivalent statements, in the sense of Proposition 2.

Other equivalent statements are obtained by interchanging left and right. In particular, given two tableaux P, Q of the same shape λ ⊢ n, there is exactly one integer ϑ_{P Q}^{−+} such that

DP \underline{QC} = ϑ_{P Q}^{−+} \underline{DC},

for every Deruyts tableau D of shape λ, with support disjoint from that of P, and for every coDeruyts tableau C of shape λ, with support disjoint from that of Q. This integer is called the (−+) symmetry coefficient of the two tableaux P and Q. The two types of symmetry coefficients satisfy the relation

ϑ_{P Q}^{−+} = ±ϑ_{QP.}^{+−}

The integers ϑ_{P Q}^{−+}(φ) are defined in an analogous way.

Given a partition λ ⊢ n, we have the infinite matrix

Θ_{t_{λ}}^{−+} = [ϑ_{P Q}^{−+}]_{P,Q∈t_{λ}} .

For every partition λ ⊢ n the (−+)symmetry coefficients of the standard tableaux satisfy the nondegenerate triangularity relations

ϑ_{P Q}^{−+} \{ \begin{matrix}= 0 & if P ≱ Q \\ ≠ 0 & if P = Q \end{matrix} .

Therefore, the matrix Θ_{st_{λ}}^{−+} is nonsingular. The entries of the inverse matrix are denoted by ϱ_{ST}^{−+}.

5.2. The Superstraightening Law. The biproduct has been defined for any pair of words of the same length; for words of different length it is set to be equal to \underline{0}. The mapping ( | ) : Mon[X] × Mon[X] → Super[X |X] which assigns to each pair of words their biproduct naturally extends to a bilinear mapping ( | ) : Super[X] × Super[X] → Super[X |X]. This mapping is equivariant with respect to the action of left and right superpolarizations:

xy (A|B) = (xy A|B), (A|B) xy = (A|B xy).

Analogously, the bitableau has been defined for any pair of tableaux of the same shape; for tableaux of different shape it is set to be equal to \underline{0}. The mapping ( | ) : (Mon[X] × Mon[X] × ⋯) × (Mon[X] × Mon[X] × ⋯) → Super[X |X] which assigns to each pair of tableaux their bitableau naturally extends to a bilinear mapping ( | ) : (Super[X] ⊗ Super[X] ⊗ ⋯) × (Super[X] ⊗ Super[X] ⊗ ⋯) → Super[X |X]. This mapping is equivariant with respect to the action of left (and right) superpolarizations:

xy ( \begin{matrix}u_{1} & v_{1} \\ ⋮ & ⋮ \\ u_{i} & v_{i} \\ ⋮ & ⋮ \\ u_{p} & v_{p} \end{matrix} ) = ∑_{i=1,...,p} ξ_{i} ( \begin{matrix}u_{1} & v_{1} \\ ⋮ & ⋮ \\ xy u_{i} & v_{i} \\ ⋮ & ⋮ \\ u_{p} & v_{p} \end{matrix} ) ,

where ξ_{i} = (−1)^{|xy||u_{1}···u_{i−1}|}.

The superalgebra Super[X] has a natural structure of bialgebra, in which the coproduct Δ(u) = ∑_{(u)} u_{(1)} ⊗ u_{(2)} of any u ∈ Super[X] is the algebra morphism defined by setting

Δ(1) = 1 ⊗ 1
Δ(x) = x ⊗ 1 + 1 ⊗ x, for all x ∈ X .

Given a positive symbol a in X , and a natural number n = 0, 1, 2, …, we set a^{(n)} = \frac{a^{n}}{n!} . Notice that a^{(n)}a^{(m)} = \binom{n+m}{n}a^{(n+m)} and Δ(a^{(n)}) = ∑_{i=0}^{n} a^{(i)} ⊗ a^{(n−i)}.

Theorem 1 (Superstraightening Law; see[26, Chapter 3, Proposition 10]). Given five words u, v, w, x, y on X , we have

∑_{(v)} ( \begin{matrix}uv_{(1)} & x \\ v_{(2)}w & y \end{matrix} ) = (−1)^{|u||v|} ∑_{(u),(y)}(−1)^{l(u_{(2)})} ( \begin{matrix}vu_{(1)} & xy_{(1)} \\ u_{(2)}w & y_{(2)} \end{matrix} ) .

For a “virtual proof”, we refer to [6, Subsection 8.1].

Remark 1. If x, y are words in negative symbols, and y is a subword of x, then we have

∑_{(v)} ( \begin{matrix}uv_{(1)} & x \\ v_{(2)}w & y \end{matrix} ) = (−1)^{|u||v|+l(u_{(2)})} ∑_{(u)} ( \begin{matrix}vu_{(1)} & x \\ u_{(2)}w & y \end{matrix} ) .

6. Properties of the Basic Objects

In this section, we state some of the main properties of the basic objects for the bimodule B_{n}[X]·Super_{n}[X |X]·[X]_{n}B; all these properties follow from the Triangularity Lemma, the Nondegeneracy Lemma and the Straightening Law, and we give a sketch of their proofs. We give references to the original papers, as well as to the closest item in [6].

6.1. Spanning, Linear Independence. For standard bitableaux, we have the following Spanning Theorem.

Theorem 2 ([26]; [6, Section 8.3]). Given a pair (P, Q) of tableaux of the same shape λ ⊢ n on X , the bitableau (P |Q) can be written as a linear combination, with rational coefficients,

(P |Q) = ∑_{S,T} c_{ST} (S|T), c_{ST} ∈ ℚ

of standard bitableaux (S|T), where (S, T) ranges through the pairs of standard tableaux, the first of the same content of P and the second of the same content of Q, with a common shape greater or equal to λ, in the dominance order.

Proof. It follows from a suitable iteration of the Straightening Law. □

Now, each monomial in Super_{n}[X |X] can be regarded as a bitableau parametrized by a pair of column-tableaux, and thus it can be written as a linear combination, with rational coefficients, of standard bitableaux with shape partition of n. This means that the set of the standard bitableaux of size n on X spans the space Super_{n}[X |X]. It can be proved, by using the Triangularity Lemma, that standard bitableaux are also linearly independent, and thus form a basis of this space.

For standard symmetrized bitableaux, we have the following Linear Independence Theorem.

Theorem 3 ([6, Subsection 9.4]). The set of all the standard symmetrized bitableaux of size n on X is linearly independent.

Proof. The basic step consists in considering the operators

\underline{ξ} ↦ DP \underline{ξ} QC, \underline{ξ} ∈ Super_{n}[X |X],

for standard tableaux P, Q, Deruyts tableau D with support disjoint from that of P, and coDeruyts tableau C with support disjoint from that of Q, all of the same shape partition of n, and proving that the action of these operators on standard symmetrized bitableaux (S| T ) of size n satisfies the nondegenerate triangularity relations

DP (S| T ) QC = \{ \begin{matrix}\underline{0} & unless P ≥ S, T ≥ Q \\ k_{λ}ϑ_{P P}^{−+} \underline{DC} ϑ_{QQ}^{−+} ≠ \underline{0} & if P = S, T = Q \end{matrix} ,

where λ in the second line is the common shape of P, S, T, Q.

This claim can be proved by writing the standard symmetrized bitableau (S| T ) in a virtual form, thus obtaining the expression

DP (S| T ) QC = DP SĊ \underline{ĊḊ} ḊT QC,

and by applying Lemma 1 (twice), Lemma 2 and Lemma 3. □

6.2. Triangularity Theorem and Orthonormal Generators.

Theorem 4 (Triangularity Theorem; see [9]; [6, Subsection 10.2]). The action of Young–Capelli symmetrizers on symmetrized bitableaux satisfies the relations

γ(P, Q ) (R| S ) = (P, Q ) (R| S )γ

= \{ \begin{matrix}k_{λ} ϑ_{QR}^{−+} (P | S ) & if sh(Q) = sh(R)(= λ) \\ \underline{0} & if sh(Q) ≠ sh(R) \end{matrix} ,

where the coefficient k_{λ} is nonzero. For Q, R standard tableaux, the symmetry coefficients ϑ_{QR}^{−+} satisfy the nondegenerate triangularity relations

ϑ_{QR}^{−+} \{ \begin{matrix}= 0 & if Q ≱ R \\ ≠ 0 & if Q = R \end{matrix} .

Proof. By writing the Young–Capelli symmetrizers and the symmetrized bitableaux in virtual form, the first equality becomes

P C CD DQ \underline{RĊ} ĊḊ ḊS = P C \underline{CD} DQ RĊ ĊḊ ḊS,

and it is readily seen as a consequence of Proposition 1. The first part of the statement follows by applying in turn Lemma 1 (twice), and Lemma 3. The second part of the statement follows by Lemma 2. □

This theorem leads to the construction of the Orthonormal Generators (see [9]; [6, Subsection 10.3]). Given two standard tableaux P, Q of the same shape λ on X , set

Y (P, Q ) = \frac{1}{k_{λ}} ∑_{A∈𝔰𝔱_{λ}[X]}ϱ_{QA}^{−+} γ(P, A ),

(R, S )Y = \frac{1}{k_{λ}} ∑_{B∈𝔰𝔱_{λ}[X]}ϱ_{BR}^{−+} (B, S )γ,

where the coefficients ϱ^{−+} are the Rutherford coefficients. Notice that in the first sum the nonzero terms are parametrized by standard tableaux A of shape λ which have the same content of Q. Thus in this sum only a finite number of nonzero terms occurs. An analogous remark holds for the other sum.

The action of the Orthonormal Generators on the standard Symmetrized Bitableaux satisfies the orthogonality relations

Y (P, Q ) (R| S ) = (P, Q ) (R| S )Y = δ_{QR} (P | S ).

7. Explicit Complete Decompositions

In this section, we revert to consider the bimodule

B_{n}[L|P] · Super_{n}[L|P] · [L|P]_{n}B.

We will give some of the main results on this bimodule, and provide a sketch of their proofs. From the remarks made in Section 4 on the expansions of virtual expressions, especially from Proposition 3, we get the following facts.

For any pair (S, T) of tableaux of the same shape partition of n on L and P, the symmetrized bitableau (S| T ) associated to (S, T) in the space Super_{n}[X |X] is a linear combination of monomials associated to pairs of words S′, T′ equivalent to S and T, and, therefore, it belongs to the space Super_{n}[L|P].

For any pair (R, S) of tableaux of the same shape partition of n on L, the left Young–Capelli symmetrizer γ(R, S ) associated to (R, S) in the algebra B_{n}[X] is a linear combination of left polarization monomials associated to pairs of words R′, S′ equivalent to subwords of R and S, and, therefore, it gives rise to an element in the algebra B_{n}[L|P], that we denote by

γ_{/}(R, S ).

Analogously, for any pair (T, U) of tableaux of the same shape partition of n on P, the right Young–Capelli symmetrizer (T, U )γ associated to (T, U) in the algebra [X]_{n}B gives rise to an element in the algebra [L|P]_{n}B, that we denote by (T, U )γ_{/}. Analogous remarks and notations are extended to each of the basic objects, and to the Orthonormal Generators.

The space Super_{n}[L|P] contains the generating set of standard bitableaux

{(S|T); (S, T) ∈ st(L) × st(P) : sh(S) = sh(T) ⊢ n} ,

as well as the linearly independent set of standard symmetrized bitableaux

{(S| T ); (S, T) ∈ st(L) × st(P) : sh(S) = sh(T) ⊢ n} .

Since the two sets have the same cardinality, they are both bases of this space: the former is called the standard basis, and the latter the Clebsch–Gordan–Capelli basis of Super_{n}[L|P].

Theorem 5 ([8, 9]; [6, Section 12]). The space Super_{n}[L|P] has the complete decomposition

Super_{n}[L|P] = _{⨁_{λ⊢n} ; λ∈H(L)∩H(P)} ⨁_{T ∈𝔰𝔱_{λ}(P)} ⟨(S| T ); S ∈ 𝔰𝔱_{λ}(L)⟩

as a semisimple left module over the algebra B_{n}[L|P]; the summands of outer direct sum are the isotypic components of the module. The algebra B_{n}[L|P] has the complete decomposition

B_{n}[L|P] = _{⨁_{λ⊢n} ; λ∈H(L)∩H(P)} ⨁_{T ∈𝔰𝔱_{λ}(L)}⟨Y_{/}(S, T ); S ∈ 𝔰𝔱_{λ}(L)⟩

as a semisimple algebra; the summands of outer direct sum are the simple subalgebras of the algebra. Analogous results hold for the space Super_{n}[L|P] as a right module over the algebra [L|P]_{n}B, and for the algebra [L|P]_{n}B.

Proof. These results follow, through arguments of elementary linear algebra, from the results of the previous section. We only remark that, for any pair (S, T) of standard tableaux of shape λ ∈ H(L) ∩ H(P) on L, the Orthonormal Generator Y_{/}(S, T ) is nonzero. Indeed, there is a standard tableau U of shape λ on P, and Y_{/}(S, T )(T, U ) =

(T, U ) ≠ \underline{0}. □

The irreducible left submodules S_{T} = ⟨(S| T ); S ∈ 𝔰𝔱_{λ}(L)⟩ that appear in the above decomposition are called Schur modules. For the irreducible representations of the algebra B_{n}[L|P] on the Schur modules, we have the following matrix form.

Proposition 4 ([6, Section 13]). For any φ ∈ B_{n}[L|P], the matrix Ξ_{T} (φ) that represents the restriction of φ to a Schur module S_{T} of type λ, with respect to the basis of standard symmetrized bitableaux is given by

Ξ_{T} (φ) = (Θ_{𝔰𝔱_{λ}(L)}^{−+}(I))^{−1} Θ_{𝔰𝔱_{λ}(L)}^{−+}(φ).

Proof. For every V, W ∈ 𝔰𝔱_{λ}(L), we have

DW (φ V C \underline{CD} DT) = DW ∑_{U∈𝔰𝔱_{λ}(L)} ξ_{T;UV} (φ) UC \underline{CD} DT,
ϑ_{W V}^{−+} (φ) DC \underline{CD} DT = ∑_{U∈𝔰𝔱_{λ}(L)} ϑ_{W U}^{−+} ξ_{T;UV} (φ) DC \underline{CD} DT,
ϑ_{W V}^{−+} (φ) = ∑_{U∈𝔰𝔱_{λ}(L)} ϑ_{W U}^{−+} ξ_{T;UV} (φ).

From these equalities, summarized in a matrix equality, we get the formula given in the statement. □

Remark 2. The space Super_{n}[L|P] has also a basis formed by monomials, e.g. by the monomials \underline{P Q} associated to pairs (P, Q) = (x_{1}x_{2} ⋯ x_{n}, y_{1}y_{2} ⋯ y_{n}) of words of length n on L, P, such that the sequence (x_{1}, y_{1}), (x_{2}, y_{2}), …, (x_{n}, y_{n}) is weakly increasing in the lexicographic order, and has no repetition of pairs of symbols of different ℤ_{2}−grade. Thus the set of these pairs of words and the set of the pairs (S, T) of standard tableaux of the same shape ⊢ n on L, P have the same cardinality. A bijection between these sets is provided by the superalgebraic extension of the Robinson–Schensted–Knuth correspondence (see [5], and [30] for further developments). This correspondence could be used in place of the straightening laws in the foundations of the theory.

8. Proofs of the Propositions on Virtual Expressions

8.1. Basic Remarks. Let (P, Q)=(p_{1} ⋯ p_{n}, q_{1} ⋯ q_{n}) and (R, S)=(r_{1} ⋯ r_{n}, s_{1} ⋯ s_{n}) be two pairs of words of length n over X , such that P and Q have disjoint supports, and consider the expression

P Q \underline{RS} = p_{1}q_{1} ⋯ p_{n}q_{n} (\underline{r_{1}s_{1}} ⋯ \underline{r_{n}s_{n}}) .

Any permutation of the elements \underline{r_{i}s_{i}} and, by our assumption, any permutation of the polarizations p_{i}q_{i} leave the expression unchanged, up to a sign.

By direct computation, we get an expansion of the form

p_{1}q_{1} ⋯ p_{n}q_{n} (\underline{r_{1}s_{1}} ⋯ \underline{r_{n}s_{n}}) = ∑_{i_{1},...,i_{n}} ϵ_{i_{1},...,i_{n}} δ_{q_{i_{1}}r_{1}} ⋯ δ_{q_{in}r_{n}} \underline{p_{i_{1}}s_{1}} ⋯ \underline{p_{i_{n}}s_{n}},

where the sum is taken over all the linear orderings i_{1}, …, i_{n} of 1, …, n and ϵ_{i_{1},...,i_{n}} ∈ {−1, +1}. This expansion can be written in the form

P Q \underline{RS} = ∑_{P′≡P} ξ_{P′} \underline{P′S},

where P′ ranges among all the words equivalent to P and the coefficients ξ_{P′} are integers. The value of the expression depends on the central pair (Q, R) only through its isomorphism type, and the expression is zero unless Q and R have the same content.

Notice that, for Q = R = S,

P Q \underline{QQ} = c_{Q}! \underline{P Q},

where c_{Q}! is the product of the factorials of the multiplicities of the symbols occurring in Q.

Analogous remarks can be made for the expressions of the form \underline{P Q} RS, where (P, Q) and (R, S) are pairs of words of length n over X , such that R and S have disjoint supports.

8.2. Proof of Proposition 1. Let (P, Q) and (R, S) be two pairs of words of length n over X , where words in the same pair have disjoint supports.

If c_{Q} ≠ c_{R}, then P Q \underline{RS} = \underline{0} = \underline{P Q} RS; if c_{Q} = c_{R}, then

\underline{P Q} RS = \frac{1}{c_{Q}!} P Q \underline{QQ} RS = \frac{1}{c_{R}!} P Q \underline{RR} RS = P Q \underline{RS}.

8.3. Proof of Proposition 2. Let

(P_{1}^{1}, Q_{1}^{1}), (P_{2}^{1}, Q_{2}^{1}), …, (P_{m_{1}}^{1}, Q_{m_{1}}^{1}), (P_{1}^{2}, Q_{1}^{2}), …, (P_{m_{2}}^{2}, Q_{m_{2}}^{2}), …, (P_{m_{u}}^{u}, Q_{m_{u}}^{u})

be a sequence of pairs of words of the same length n on X , such that in each pair the words have disjoint supports. Let ϑ_{1}, ϑ_{2}, …, ϑ_{u} be scalars, and let j_{1}, j_{2}, …, j_{u} be indexes with 1 ≤ j_{i} ≤ m_{i}. We consider the equalities

∑_{i} ϑ_{i} \underrightarrow{P_{1}^{i}Q_{1}^{i}} \underrightarrow{P_{2}^{i}Q_{2}^{i}} ⋯ \underrightarrow{P_{m_{i}}^{i}Q_{m_{i}}^{i}} |_{n} = \underrightarrow{0}|_{n} ,

∑_{i} ϑ_{i} P_{1}^{i}Q_{1}^{i} ⋯ P_{j_{i}−1}^{i}Q_{j_{i}−1}^{i} \underline{P_{j_{i}}^{i}Q_{j_{i}}^{i}} P_{j_{i}+1}^{i}Q_{j_{i}+1}^{i} ⋯ P_{m_{i}}^{i}Q_{m_{i}}^{i} = \underline{0},

∑_{i} ϑ_{i} \underleftarrow{P_{1}^{i}Q_{1}^{i}} \underleftarrow{P_{2}^{i}Q_{2}^{i}} ⋯ \underleftarrow{P_{m_{i}}^{i}Q_{m_{i}}^{i}} |_{n} = \underleftarrow{0}|_{n} .

Any two equalities of the second type are equivalent; indeed, by Proposition 1, their left-hand sides are the same. We can assume that all the rightmost tableaux Q_{m_{1}}^{1}, Q_{m_{2}}^{2}, …, Q_{m_{u}}^{u} in the various monomials have the same content, let Q be one of them.

We prove the equivalence of the first equality with one equality of the second type.

First we prove that the first equality implies the second. Indeed,

∑ ϑ_{i} P_{1}^{i}Q_{1}^{i} ⋯ P_{m_{i}−1}^{i}Q_{m_{i}−1}^{i} \underline{P_{m_{i}}^{i}Q_{m_{i}}^{i}}

= \frac{1}{c_{Q}!} ∑ ϑ_{i} P_{1}^{i}Q_{1}^{i} ⋯ P_{m_{i}−1}^{i}Q_{m_{i}−1}^{i} P_{m_{i}}^{i}Q_{m_{i}}^{i} \underline{QQ} = \underline{0}.

Now we prove that the first equality is implied by the second. Indeed, if (R, S) is any pair of words of length n, and Ṙ is an isomorphic copy of R with support disjoint from the supports of R and S, then

∑ ϑ_{i} P_{1}^{i}Q_{1}^{i} ⋯ P_{m_{i}}^{i}Q_{m_{i}}^{i} \underline{RS}

= \frac{1}{c_{R}!} ∑ ϑ_{i} P_{1}^{i}Q_{1}^{i} ⋯ P_{m_{i}}^{i}Q_{m_{i}}^{i} \underline{RṘ} ṘS

= \frac{1}{c_{R}!} ∑ ϑ_{i} P_{1}^{i}Q_{1}^{i} ⋯ \underline{P_{m_{i}}^{i}Q_{m_{i}}^{i}} RṘ ṘS = \underline{0}.

8.4. Proof of Proposition 3. In this subsection, for any word A on X , we denote by X − A the complement of the support of A in X , and use the symbol |_{X −A} to mean restriction to the subalgebra Super[X − A|X] of the algebra Super[X |X].

Step 1. Let P, Q be two words of the same length n on X . There is a family of integers (η_{P′,Q′}) indexed by the pairs of words P′, Q′ of the same length equivalent to subwords of P and Q, such that, for every word A of length n with pairwise distinct positive entries occurring neither in P nor in Q,

\underrightarrow{P A} \underrightarrow{AQ}|_{X −A} = ∑_{P′,Q′} η_{P′,Q′} \underrightarrow{P′Q′}|_{X −A} .

We use induction on the common length n ≥ 1 of the words P and Q. For n = 1 the statement is true; assume n > 1.

Let P = p_{1}p_{2} ⋯ p_{n}, Q = q_{1}q_{2} ⋯ q_{n}, and A = a_{1}a_{2} ⋯ a_{n}, where each symbol a_{i} is distinct by any other symbol a_{j}, p_{h}, q_{k}. Up to a sign, we have

\underrightarrow{P A} \underrightarrow{AQ} = p_{n}a_{n} ⋯ p_{2}a_{2} p_{1}a_{1} a_{1}q_{1} a_{2}q_{2} ⋯ a_{n}q_{n},

where, for the sake of brevity, we have denoted left polarizations without under-arrows.

By applying the commutation relation to the central factors p_{1}a_{1} and a_{1}q_{1} we get

(−1)^{|p_{1}||q_{1}|} p_{n}a_{n} ⋯ p_{2}a_{2} (a_{1}q_{1} p_{1}a_{1}) a_{2}q_{2} ⋯ a_{n}q_{n} + p_{n}a_{n} ⋯ p_{2}a_{2} (p_{1}q_{1}) a_{2}q_{2} ⋯ a_{n}q_{n}

+ (−1)^{|p_{1}||q_{1}|} δ_{p_{1}q_{1}} p_{n}a_{n} ⋯ p_{2}a_{2} (a_{1}a_{1}) a_{2}q_{2} ⋯ a_{n}q_{n}.

The first and the last term of this sum of operators vanish on the subalgebra Super[X − A|X]. By applying the commutation relations to the second term we get

(−1)^{|p_{1}q_{1}||q_{2}···q_{u}|} p_{n}a_{n} ⋯ p_{2}a_{2} a_{2}q_{2} ⋯ a_{n}q_{n} p_{1}q_{1}

+ ∑_{i=2}^{n}(−1)^{|p_{1}q_{1}||q_{2}···q_{i−1}|} δ_{p_{1}q_{i}} p_{n}a_{n} ⋯ p_{2}a_{2} a_{2}q_{2} ⋯ a_{i−1}q_{i−1} a_{i}q_{1} a_{i+1}q_{i+1} ⋯ a_{n}q_{n}.

Now the statement follows by applying the induction hypothesis to the pair P_{1} = p_{n} ⋯ p_{2}, Q_{1} = q_{2} ⋯ q_{n} and to the pairs P_{1} = p_{n} ⋯ p_{2}, Q_{i} = q_{2} ⋯ q_{i−1}q_{1}q_{i+1} ⋯ q_{n}.

Step 2. Under the same notation of the previous step, we have

\underrightarrow{P A} \underrightarrow{AQ}|_{n} = ∑_{P′,Q′} η_{P′,Q′} \underrightarrow{P′Q′}|_{n} .

Let \underline{RS} be a monomial in Super_{n}[X |X]. Then, if B is a word of length n with pairwise distinct positive entries occurring neither in P nor in Q nor in R, we have

P A AQ \underline{RS} = P B BQ \underline{RS} = [ ∑_{P′,Q′} η_{P′,Q′}P′Q′] \underline{RS}.

This means that

\underrightarrow{P T} \underrightarrow{T Q}|_{n} = ∑_{P′,Q′} η_{P′,Q′} \underrightarrow{P′Q′}|_{n} .

Step 3. Let (P_{1}, Q_{1}), (P_{2}, Q_{2}), …, (P_{u}, Q_{u}) be any sequence of pairs of words of the same length n on X , such that in any pair the words have disjoint supports.

In the space Super_{n}[X |X] we have an expansion of the form

P_{1}Q_{1} ⋯ P_{u−1}Q_{u−1} \underline{P_{u}Q_{u}} = ∑_{P′≡P} ξ_{P′} \underline{P′Q_{u}} = ∑_{P′≡P} ζ_{P′} P′T \underline{T Q_{u}},

where P′ ranges among the words equivalent to P_{1}, the coefficients ξ_{P′} are integers, T is a word with pairwise distinct entries which do not occur neither in P_{1} nor in Q_{u}, and each coefficient ζ_{P′} equals, up to a sign, the corresponding coefficient ξ_{P′}. From Proposition 2, this expansion can be lifted to the expansion

\underrightarrow{P_{1}Q_{1}} ⋯ \underrightarrow{P_{u−1}Q_{u−1}} \underrightarrow{P_{u}Q_{u}} |_{n} = ∑_{P′≡P} ζ_{P′} \underrightarrow{P′T} \underrightarrow{T Q_{u}}, |_{n}

in B_{n}[X]. By applying the result of previous step to each summand, we get the proof of Proposition 3.

9. Proof of the Triangularity Lemma

9.1. Proof of the Scholion. Let (A, B) be a pair of words of the same length on X , such that A has the same content of a coDeruyts tableau C, and B has the same content of a Deruyts tableau D. We set

C = a_{1}^{λ_{1}} ; a_{2}^{λ_{2}} ; ⋮ ; a_{p}^{λ_{p}} , D = b_{1} · · · · · · · · · b_{μ_{1}} ; b_{1} · · · · · · b_{μ_{2}} ; ⋮ ; b_{1} ⋯ b_{μ_{p}} ,

where the a_{i} are pairwise distinct positive symbols, the b_{j} are pairwise distinct negative symbols, and ∑ λ_{i} = ∑ μ_{j}. We prove that, under the assumption \underline{AB} ≠ 0, we have

(1) sh(C) ≤ sh(D);

(2) sh(C) = sh(D) ⇒ \underline{AB} = η \underline{CD}, where η ∈ \{±1\}.

The variables \underline{a_{i}b_{j}} are negative; since \underline{AB} ≠ \underline{0}, each of these variables appears in this monomial at most once. Let M be the matrix, with rows indexed by a_{1}, a_{2}, … and columns indexed by b_{1}, b_{2}, …, whose (a_{i}, b_{j})−th entry is 1 if the variable \underline{a_{i}b_{j}} appears in the monomial \underline{AB}, and 0 otherwise; the rows of M sum up to (λ_{1}, λ_{2}, …) = λ = sh(C). Let M′ be the matrix obtained from M by migrating the 1′s upward columnwise; the rows of M′ sum up to (μ_{1}, μ_{2}, …) = μ = sh(D).

Now, for every h = 1, 2, …, we have

λ_{1} + λ_{2} + … λ_{h} ≤ μ_{1} + μ_{2} + … μ_{h}.

Indeed, the left-hand side counts the 1′s in the first h rows of M; these 1′s appear also in the first h rows of M′, and the total number of 1′s in the first h rows of M′ is counted by the right-hand side.

If λ = μ, then no upward migration takes place, so the 1′s in the i−th row have the column index j ranging from 1 to λ_{i}, and the monomial is

\underline{AB} = ±\underline{a_{1}b_{1}} ⋯ \underline{a_{1}b_{λ_{1}}} \underline{a_{2}b_{1}} ⋯ \underline{a_{2}b_{λ_{2}}} ⋯ = ±\underline{CD}.

9.2. Proof of the Triangularity Lemma I. Let (A_{1}, B_{1}), (A_{2}, B_{2}), …, (A_{s}, B_{s}) be pairs of words of the same length on X , where any two words in the same pair have disjoint supports, A_{1} has the same content of a coDeruyts tableau C and B_{s} has the same content of a Deruyts tableau D. Then we have

A_{1}B_{1} ⋯ \underline{A_{s}B_{s}} = ∑_{A≡A_{1}} ξ_{A} \underline{AB_{s}},

where A ranges among the words equivalent to the word A_{1}, and hence equivalent to C, and the coefficients ξ_{A} are integers. By the Scholion,

• if sh(C) ≰ sh(D), then for each A we have \underline{AB_{s}} = \underline{0}, thus

A_{1}B_{1} ⋯ \underline{A_{s}B_{s}} = \underline{0};

• if sh(C) = sh(D), then for each A we have \underline{AB_{s}} = η_{A} \underline{CD}, with η_{A} ∈ \{0, ±1\}, thus

A_{1}B_{1} ⋯ \underline{A_{s}B_{s}} = ϑ \underline{CD},

for some ϑ ∈ ℤ.

9.3. Expansions of Biproducts. The expansions of the biproducts

(P |Q) = (p_{1} ⋯ p_{n}|q_{1} ⋯ q_{n}),

where all the symbols in P have the same ℤ_{2}−degree and all the symbols in Q have the same ℤ_{2}−degree, can be arranged in the following table:

\begin{matrix}c_{P}! ∑_{P′≡P} \underline{P′Q} = c_{Q}! ∑_{Q′≡Q} \underline{P Q′} & c_{P}! ∑_{P′≡P} \underline{P′Q} = ∑_{Q′≡Q} ϵ_{Q,Q′} \underline{P Q′} \\ ∑_{P′≡P} ϵ_{P,P′} \underline{P′Q} = c_{Q}! ∑_{Q′≡Q} \underline{P Q′} & (−1)^{\binom{n}{2}} ∑_{P′≡P} ϵ_{P,P′} \underline{P′Q} = (−1)^{\binom{n}{2}} ∑_{Q′≡Q} ϵ_{Q,Q′} \underline{P Q′} \end{matrix}

The rows and the columns of this 2×2 table are indexed by 0, 1; the (i, j)−the entry of the table displays the left and right expansions of the biproduct (P |Q) when all the entries in P have ℤ_{2}−degree i and all the entries in Q have ℤ_{2}−degree j. When all the symbols in one of the two words, say P, are negative, we assume that P is square-free; if P′ ≡ P is a word equivalent to P, then the sign of any permutation which carries P to P′ equals the sign ϵ_{P,P′}, defined by the relation P = ϵ_{P,P′} P′ in Super[X].

The table of the expansions of the ∗−biproducts

(P |Q)^{∗} = (p_{1} ⋯ p_{n}|q_{1} ⋯ q_{n})^{∗},

where all the symbols in P have the same ℤ_{2}−degree and all the symbols in Q have the same ℤ_{2}−degree, is the transpose of the above table.

9.4. Proof of the Triangularity Lemma II. Let (C, S), (T, D) be pairs of tableaux of the same size, where tableaux in the same pair have the same shape and disjoint supports, C is coDeruyts, S and T are standard, and D is Deruyts. We set

C = (α_{i})_{(i,j)∈λ}, S = (s_{ij})_{(i,j)∈λ} T = (t_{hk})_{(h,k)∈μ}, D = (β_{k})_{(h,k)∈μ},

with λ, μ ⊢ n, where none of the α′s equals any of the s′s, and none of the t′s equals any of the β′s.

Proof of the first part of the statement. Assume that S ≰ T. If c_{S} ≠ c_{T} , then CS \underline{T D} = \underline{0}. If c_{S} = c_{T} , then the assumption S ≰ T means that for some letter x ∈ X , the shape γ of the subtableau (s_{ij} : s_{ij} ≤ x) of the tableau S is not less than or equal, in the dominance order, to the shape δ of the subtableau (t_{hk} : t_{hk} ≤ x) of the tableau T. In the following, the symbol of a tableau with a partition as subscript will denote its subtableau of shape that partition. Then we have

CS \underline{T D} = ± ∏_{s_{ij}≤x} α_{i}s_{ij} ∏_{s_{ij}>x} α_{i}s_{ij} ( ∏_{t_{hk}≤x} \underline{t_{hk}β_{k}} ∏_{t_{hk}>x} \underline{t_{hk}β_{k}} )

= ± ( ∏_{s_{ij}≤x} α_{i}s_{ij} ∏_{t_{hk}≤x} \underline{t_{hk}β_{k}} ) ( ∏_{s_{ij}>x} α_{i}s_{ij} ∏_{t_{hk}>x} \underline{t_{hk}β_{k}} )

= ± (C_{γ}S_{γ} \underline{T_{δ}D_{δ}}) ( ∏_{s_{ij}>x} α_{i}s_{ij} ∏_{t_{hk}>x} \underline{t_{hk}β_{k}} )

= 0,

by Lemma 1.

Proof of the second part of the statement. Let S = (s_{ij})_{(i,j)∈λ} = T, with λ ⊢ n.

We use induction on the number of distinct letters which appear in S. For one letter, the statement is true: for |x| = 0 we have λ = (n), and

CS \underline{SD} = α_{1} … α_{1}x … x \underline{x … xβ_{1} … β_{n}} = n! \underline{α_{1} … α_{1}β_{1} … β_{n}} = n! \underline{CD};

for |x| = 1 we have λ = (1, 1, …, 1), and

CS \underline{SD} = α_{1} ; ⋮ ; α_{n} x ; ⋮ ; x \underline{x ; ⋮ ; x β_{1} ; ⋮ ; β_{n}} = n! \underline{CD}.

We assume now that S contains more than one letter; we denote by x the greatest letter occurring in S, and denote by γ the shape of the subtableau (s_{ij} : s_{ij} < x) of the tableau S.

We consider the case in which x is positive (the case x negative is analogous). The cells of the shape λ in which x occurs form an horizontal strip; listing these cells starting from the leftmost and moving right and up, we get a sequence

(i_{1}, j_{1}), (i_{2}, j_{2}), …, (i_{m}, j_{m})

where the sequence of the row indexes is of the form

i_{1} = … = i_{m_{1}} > i_{m_{1}+1} = … = i_{m_{1}+m_{2}} > … > i_{m−m_{r}} = … = i_{m}

and the sequence of the column indexes is strictly increasing.

We have

CS \underline{SD} = ± ( ∏_{(i,j)∈γ} α_{i}s_{ij}) α_{i_{1}}x … α_{i_{m}}x (( ∏_{(i,j)∈γ} \underline{s_{ij}β_{j}} ) \underline{xβ_{j_{1}}} ⋯ \underline{xβ_{j_{m}}} )

= ± ( ∏_{(i,j)∈γ} α_{i}s_{ij} ∏_{(i,j)∈γ} \underline{s_{ij}β_{j}} ) α_{i_{1}}x … α_{i_{m}}x \underline{xβ_{j_{1}}} ⋯ \underline{xβ_{j_{m}}}

= ± (θ_{S_{γ}S_{γ}}^{+−} ∏_{(i,j)∈γ} \underline{α_{i}β_{j}} ) m_{1}!m_{2}! ⋯ m_{s}! ∑ \underline{α_{i′_{1}}β_{j_{1}}} ⋯ \underline{α_{i′_{m}}β_{j_{m}}},

where the sum is extended to the words i′_{1} ⋯ i′_{m} equivalent to the word i_{1} ⋯ i_{m}.

We claim that i_{1} ⋯ i_{m} is the only word which really contributes to the whole expression. Indeed, for each cell (i_{p}, j_{p}) in the horizontal strip in which occurs x, any cell (i′_{p}, j_{p}), with i′_{p} < i_{p}, belongs to the shape γ, so

( ∏_{(i,j)∈γ} \underline{α_{i}β_{j}}) \underline{α_{i′_{p}}β_{j_{p}}} = 0.

Thus the words i′_{1} ⋯ i′_{m} whose corresponding summand contributes to the whole expression are subject to the conditions i′_{p} ≥ i_{p}, for all p = 1, …, m, and there is only one such word: i_{1} ⋯ i_{p}.

Now we have

CS \underline{SD} = ±m_{1}!m_{2}! ⋯ m_{s}! θ_{S_{γ}S_{γ}}^{+−} ( ∏_{(i,j)∈γ} \underline{α_{i}β_{j}}) \underline{α_{i_{1}}β_{j_{1}}} ⋯ \underline{α_{i_{m}}β_{j_{m}}},

= ±m_{1}!m_{2}! ⋯ m_{s}! θ_{S_{γ}S_{γ}}^{+−} ∏_{(ij)∈λ} \underline{α_{i}β_{j}}

= ±m_{1}!m_{2}! ⋯ m_{s}! θ_{S_{γ}S_{γ}}^{+−} \underline{CD}.

Finally, by induction, we get

θ_{SS}^{+−} = ±m_{1}!m_{2}! ⋯ m_{s}! θ_{S_{γ}S_{γ}}^{+−} = ±m_{1}!m_{2}! ⋯ m_{s}! c_{S_{γ}}^{r}! c_{S_{γ}}^{c}! = ±c_{S}^{r}! c_{S}^{c}!.

10. Proofs of the Nondegeneracy Lemma

10.1. Expansions of Bitableaux. Let P, Q be tableaux of the same shape λ ⊢ n on X , and consider the bitableau

P C \underline{CQ} = \underline{P C} CQ = (P |Q),

where C is a coDeruyts tableau of shape λ with no symbol in common with P and Q.

Notice that, upon denoting by P_{1}, P_{2}, …, P_{m} the rows of P, by A_{1}, A_{2}, …, A_{m} the rows of C, by Q_{1}, Q_{2}, …, Q_{m} the rows of Q, we have

(P |Q) = P C \underline{CQ} = ∏_{i=1}^{m} P_{i}A_{i} ∏_{i=1}^{m} \underline{A_{i}Q_{i}}

= (−1)^{∑_{i>j} |P_{i}||Q_{j}|} ∏_{i=1}^{m} (P_{i}A_{i} \underline{A_{i}Q_{i}}) = (−1)^{∑_{i>j} |P_{i}||Q_{j}|} ∏_{i=1}^{m} (P_{i}|Q_{i});

thus, the bitableau of the tableaux P and Q is, up to a sign, the product of the biproducts of the corresponding rows of P and Q.

Among the various left and right expansions of the bitableau (P |Q), we consider only the right expansion in the case in which all the symbols in P and Q are negative; we assume that each row of P and each row of Q is square-free. In this case we have

(P |Q) = (−1)^{\binom{n}{2}} ∑_{Q′≡_{r}Q} ϵ_{Q,Q′} \underline{P Q′},

where Q′ runs through the tableaux row-equivalent to Q, and ϵ_{Q,Q′} is the sign of the permutations which transform Q to Q′; notice that, if Q′ is any tableau of the same shape as Q, then

Q ≡_{r} Q′ if and only if CQ = ϵ_{Q,Q′} CQ′.

Indeed,

(P |Q) = (−1)^{∑_{i>j} λ_{i}λ_{j}} ∏_{i=1}^{m} (P_{i}|Q_{i})

= (−1)^{∑_{i>j} λ_{i}λ_{j}} ∏_{i=1}^{m} (−1)^{\binom{λ_{i}}{2}} ∑_{Q′_{i}≡Q_{i}} ϵ_{Q_{i},Q′_{i}} \underline{P_{i}Q′_{i}}

= (−1)^{∑_{i>j} λ_{i}λ_{j}+∑_{i} \binom{λ_{i}}{2}} ∑_{Q′_{1},Q′_{2},...≡Q_{1},Q_{2},...} ∏_{i=1}^{m} ϵ_{Q_{i},Q′_{i}} \underline{P_{i}Q′_{i}}

= (−1)^{\binom{n}{2}} ∑_{Q′≡_{r}Q} ϵ_{Q,Q′} \underline{P Q′}.

The equality ∑_{i>j} λ_{i}λ_{j} + ∑_{i} \binom{λ_{i}}{2} = \binom{n}{2}, comes from the fact that the left hand side counts the number of inversions of the permutation n(n − 1) … 21 by considering it as split into blocks of order (λ_{1}, λ_{2}, …, λ_{m}).

Let P, Q be tableaux of the same shape λ ⊢ n on X , and consider the ∗−bitableau

P D \underline{DQ} = \underline{P D} DQ = (P |Q)^{∗},

where D is a Deruyts tableau of shape λ with no symbol in common with P and Q.

We point out that, among the various expansions, in the case in which all the symbols in Q are negative, we have the right expansion

(P |Q)^{∗} = \underline{P D} DQ = c_{Q}^{c}! ∑_{Q′≡_{c}Q} \underline{P Q′},

where Q′ runs through the tableaux column-equivalent to Q.

10.2. Proof of the Nondegeneracy Lemma — Weak Form. Let C, Ḋ, Ċ, D be coDeruyts and Deruyts tableaux, all of the same shape λ ⊢ n. By Lemma 1, we have

CḊ \underline{ḊĊ} ĊD = k_{λ} \underline{CD},

for some integer k_{λ}. We prove that k_{λ} ≠ 0.

Step 1. We have

\underline{ḊĊ} ĊD = (−1)^{(_{2}^{n})} ∑_{E≡_{r}D} ϵ_{E,D} \underline{ḊE},

where the sum is taken over all the tableaux E which are row-equivalent to the tableau D, and the coefficient ϵ_{E,D} is defined by the relation

\underline{ĊE} = ϵ_{E,D} \underline{ĊD}.

Step 2. For every tableau E ≡_{r} D, we have

CḊ \underline{ḊE} = \underline{CḊ} ḊE = c_{E}^{c}! ∑_{F ≡_{c}E} \underline{CF} ,

where the sum is taken over all the tableaux F which are column-equivalent to the tableau E.

Step 3. By the previous steps, we have

CḊ \underline{ḊĊ} ĊD = (−1)^{(_{2}^{n})} ∑_{E≡_{r}D} ϵ_{E,D} CḊ \underline{ḊE}

= (−1)^{(_{2}^{n})} ∑_{E≡_{r}D} ϵ_{E,D} c_{E}^{c}! ∑_{F ≡_{c}E} \underline{CF} .

Since F must be equivalent to D, we have that \underline{CF} = \underline{0} unless F ≡_{r} D; in the case F ≡_{r} D, we have

\underline{CF} = ϵ_{F,D} \underline{CD}.

Then we have

CḊ \underline{ḊĊ} ĊD = (−1)^{(_{2}^{n})} ( ∑_{E≡_{r}D} ϵ_{E,D} c_{E}^{c}!_{∑_{F ≡_{c}E} ; F ≡_{r}D}ϵ_{F,D}) \underline{CD}.

Now, the set

{G | G ≡_{r} D}

of the tableau which are row-equivalent to the tableau D splits, under the equivalence relation ≡_{c} of column-equivalence, into some classes C_{0}, C_{1}, …; so, noting that the coefficient c_{E}^{c}! is constant over each class and denoting by γ_{i} its value on C_{i}, we get

∑_{E≡_{r}D} ϵ_{E,D} c_{E}^{c}!_{∑_{F ≡_{c}E} ; F ≡_{r}D}ϵ_{F,D} = ∑_{i} γ_{i} ∑_{E∈C_{i}} ϵ_{E,D} _{∑_{F ≡_{c}E} ; F ≡_{r}D}ϵ_{F,D}

= ∑_{i} γ_{i} ∑_{E∈C_{i}} ϵ_{E,D} ∑_{F ∈C_{i}} ϵ_{F,D} = ∑_{i} γ_{i} (∑_{E∈C_{i}} ϵ_{E,D})^{2} ≥ 0.

As a matter of fact, this integer is strictly positive, since for the equivalence class, say C_{0}, of the tableau D we have

∑_{E∈C_{0}} ϵ_{E,D} = ϵ_{D,D} = 1.

10.3. An Application of the Straightening Law. We will use the following application of the Straightening Law.

Proposition 5. Let x, …, y, z be j pairwise distinct negative letters, let l_{1} ≥ l_{2} ≥ ⋯ ≥ l_{r}, and let D be any Deruyts tableau of shape (j + l_{1}, j + l_{2}, …, j + l_{r}).

• Let a_{1}, a_{2}, …, a_{r} be pairwise distinct positive letters. Then

a_{1}z ( x ⋯ yza_{1}^{(l_{1})} ; x ⋯ yza_{2}^{(l_{2})} D ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) = (−1)^{j−1}(l_{1} + r) ( x ⋯ ya_{1}^{(l_{1}+1)} ; x ⋯ yza_{2}^{(l_{2})} D ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) .

• Let a_{1} be a positive symbol and let v_{i}, for i = 2, …, r, be words of length l_{i} not containing z. Then

a_{1}z ( x ⋯ yza_{1}^{(l_{1})} ; x ⋯ yzv_{2} D ; ⋮ ; x ⋯ yzv_{r} ) = (−1)^{j−1}(l_{1} + r) ( x ⋯ ya_{1}^{(l_{1}+1)} ; x ⋯ yzv_{2} D ; ⋮ ; x ⋯ yzv_{r} ) .

Notice that l_{1} + r is the hook length of the cell (1, j) in the Ferrers diagram of the common shape (j + l_{1}, j + l_{2}, …, j + l_{r}) of the tableaux.

Proof. We prove the first statement of the Proposition; the second follows from the first by acting on both sides with the left polarization monomial

v_{2}a_{2}^{l_{2}} ⋯ v_{r}a_{r}^{l_{r}}.

We notice that

a_{1}z ( x ⋯ yza_{1}^{(l_{1})} ; x ⋯ yza_{2}^{(l_{2})} D ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) = ∑_{i=1}^{r}(−1)^{ij−1} ( x ⋯ yza_{1}^{(l_{1})} ; ⋮ ; x ⋯ ya_{1}a_{i}^{(l_{i})} D ; x ⋯ yza_{i+1}^{(l_{i+1})} ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) .

The first statement of the proposition follows from the following facts.

ON THE COMBINATORICS OF YOUNG–CAPPELLI SYMMETRIZERS 31

(1) The first term of the sum is

(−1)^{j−1} ( x ⋯ ya_{1}a_{1}^{(l_{1})} ; x ⋯ yza_{2}^{(l_{2})} D ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) = (−1)^{j−1}(l_{1} + 1) ( x ⋯ ya_{1}^{(l_{1}+1)} ; x ⋯ yza_{2}^{(l_{2})} D ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) .

(2) The second term equals

− ( x ⋯ yza_{1}^{(l_{1})} ; x ⋯ ya_{1}a_{2}^{(l_{2})} ; x ⋯ yza_{3}^{(l_{3})} D ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) = (−1)^{j−1} ( x ⋯ ya_{1}^{(l_{1}+1)} ; x ⋯ yza_{2}^{(l_{2})} D ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) .

Indeed, by the straightening law relative to the diagonalization of a_{1}^{(l_{1}+1)}, we have

\binom{x ⋯ yza_{1}^{(l_{1})} D′}{x ⋯ ya_{1}a_{2}^{(l_{2})}} = − \binom{x ⋯ ya_{1}^{(l_{1}+1)} D′}{zx ⋯ ya_{2}^{(l_{2})}} = (−1)^{j} \binom{x ⋯ ya_{1}^{(l_{1}+1)} D′}{x ⋯ yza_{2}^{(l_{2})}} .

(3) All the terms from the second on are equal:

(−1)^{ij−1} ( x ⋯ yza_{1}^{(l_{1})} ; ⋮ ; x ⋯ ya_{1}a_{i}^{(l_{i})} D ; x ⋯ yza_{i+1}^{(l_{i+1})} ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) = (−1)^{(i−1)j−1} ( x ⋯ yza_{1}^{(l_{1})} ; ⋮ ; x ⋯ ya_{1}a_{i−1}^{(l_{i−1})} D ; x ⋯ yza_{i}^{(l_{i})} ; ⋮ ; x ⋯ yza_{r}^{(l_{r})} ) ,

for i = 2, …, r.

Indeed, by the straightening law relative to the diagonalization of a_{1}, we have

( ⋮ ; x ⋯ yza_{i−1}^{(l_{i−1})} D′′ ; x ⋯ ya_{1}a_{i}^{(l_{i})} ) = − ( ⋮ ; x ⋯ ya_{1}a_{i−1}^{(l_{i−1})} D′′ ; zx ⋯ ya_{i}^{(l_{i})} ) − ( ⋮ ; x ⋯ yza_{1}a_{i−1}^{(l_{i−1}−1)} D′′ ; a_{i−1}x ⋯ ya_{i}^{(l_{i})} )

= (−1)^{j} ( ⋮ ; x ⋯ ya_{1}a_{i−1}^{(l_{i−1})} D′′ ; x ⋯ yza_{i}^{(l_{i})} );

the second term in the first equality vanishes since it has the factor

\binom{x ⋯ yza_{1}^{(l_{1})} D′′′}{x ⋯ yza_{1}a_{i−1}^{(l_{i−1}−1)}} ,

which in turn vanishes, by the straightening law relative to the diagonalization of a_{1}^{(l_{1}+1)}.

□

32 FRANCESCO REGONATI

10.4. Proof of the Nondegeneracy Lemma. Let λ = (λ_{1}, λ_{2}, …, λ_{p}) ⊢ n be a partition of n, and let

h_{ij} = λ_{i} + λ′_{j} − i − j + 1

be the hook length of the cell (i, j) in the shape λ.

Let a_{1}, a_{2}, …, a_{p} be positive pairwise distinct symbols, let x_{1}, x_{2}, ..., x_{λ_{1}} be negative pairwise distinct symbols, and let D be any Deruyts tableau of shape λ.

From the second point of the previous proposition we get

a_{1}x_{j} ( x_{1} · · · · · · x_{j}a_{1}^{(λ_{1}−j)} ; x_{1} · · · · · · x_{λ_{2}}  D ; ⋮ ; x_{1} ⋯ x_{λ_{p}} ) = (−1)^{j−1}h_{1j} ( x_{1} · · · · · · x_{j−1}a_{1}^{(λ_{1}−j+1)} ; x_{1} · · · · · · x_{λ_{2}}  D ; ⋮ ; x_{1} ⋯ x_{λ_{p}} ) ,

for every j = 1, …, λ_{1}. By iterating this identity, we get

(∏_{j=1}^{λ_{1}} a_{1}x_{j}) ( x_{1} · · · · · · · · · x_{λ_{1}} ; x_{1} · · · · · · x_{λ_{2}} D ; ⋮ ; x_{1} ⋯ x_{λ_{p}} ) = (−1)^{\binom{λ_{1}}{2}} (∏_{j=1}^{λ_{1}} h_{1j}) ( a_{1}^{(λ_{1})} ; x_{1} · · · · · · x_{λ_{2}} D ; ⋮ ; x_{1} ⋯ x_{λ_{p}} ) .

By iterating this identity, we get

(∏_{i=1}^{p} ∏_{j=1}^{λ_{i}} a_{i}x_{j}) ( x_{1} · · · · · · · · · x_{λ_{1}} ; x_{1} · · · · · · x_{λ_{2}} D ; ⋮ ; x_{1} ⋯ x_{λ_{p}} ) = ϵ (∏_{i=1}^{p} ∏_{j=1}^{λ_{i}} h_{i,j}) ( a_{1}^{(λ_{1})} ; a_{2}^{(λ_{2})} D ; ⋮ ; a_{p}^{(λ_{p})} ) ,

where ϵ = (−1)^{∑_{h>k} λ_{h}λ_{k}+∑_{h} \binom{λ_{h}}{2}} = (−1)^{\binom{n}{2}}.

Now, for

C = a_{1}^{λ_{1}} ; a_{2}^{λ_{2}} ; ⋮ ; a_{p}^{λ_{p}} ,  Ḋ = x_{1} · · · · · · · · · x_{λ_{1}} ; x_{1} · · · · · · x_{λ_{2}} ; ⋮ ; x_{1} ⋯ x_{λ_{p}}

and Ċ any coDeruyts tableau of shape λ, we get

CḊ (\underline{ḊĊ} ĊD) = (∏_{i=1}^{p} ∏_{j=1}^{λ_{i}} a_{i}x_{j}) ( x_{1} · · · · · · · · · x_{λ_{1}} ; x_{1} · · · · · · x_{λ_{2}}  D ; ⋮ ; x_{1} ⋯ x_{λ_{p}} )

= ϵ (∏_{i=1}^{p} ∏_{j=1}^{λ_{i}} h_{i,j}) ( a_{1}^{(λ_{1})} ; a_{2}^{(λ_{2})} D ; ⋮ ; a_{p}^{(λ_{p})} ) = (−1)^{\binom{n}{2}}h_{λ}\underline{CD},

which is the statement of Proposition 5.

Remark 3. From the second identity in the proof, we get

[x_{λ_{1}} ⋯ x_{1}|x_{1} ⋯ x_{λ_{1}}] ( \begin{matrix}x_{1} · · · · · · · · · x_{λ_{1}} & y_{1} · · · · · · · · · y_{λ_{1}} \\ x_{1} · · · · · · x_{λ_{2}} & y_{1} · · · · · · y_{λ_{2}} \\ ⋮ & ⋮ \\ x_{1} ⋯ x_{λ_{p}} & y_{1} ⋯ y_{λ_{p}} \end{matrix} )

= (∏_{j=1}^{λ_{1}} h_{1j}) ( \begin{matrix}x_{1} · · · · · · · · · x_{λ_{1}} & y_{1} · · · · · · · · · y_{λ_{1}} \\ x_{1} · · · · · · x_{λ_{2}} & y_{1} · · · · · · y_{λ_{2}} \\ ⋮ & ⋮ \\ x_{1} ⋯ x_{λ_{p}} & y_{1} ⋯ y_{λ_{p}} \end{matrix} ) .

This identity in turn, for λ_{1} = λ_{2} = … = λ_{p}(= m), yields

[x_{m} ⋯ x_{1}|x_{1} ⋯ x_{m}] (x_{1} ⋯ x_{m}|x_{1} ⋯ x_{m})^{p} = ⟨p⟩_{m} (x_{1} ⋯ x_{m}|x_{1} ⋯ x_{m})^{p} ,

where ⟨p⟩_{m} = p(p + 1) … (p + m − 1). This identity is equivalent to the Cayley identity for the action of the Ω operator on a product of determinants [38].

Appendix: An Algebra of Virtual Expressions

Given a ℤ_{2}-graded set X consisting of countably many positive and countably many negative symbols, and given a positive integer n, we define a new associative algebra over the rational numbers ℚ.

First, consider the set of all pairs (P, Q) of words, of length n with disjoint supports, over X; for the sake of brevity, we write P Q instead of (P, Q). This set generates a free unitary associative algebra T over ℚ; we denote the product of two generators P Q and RS by juxtaposition P Q RS, with an inner blank; the products A_{1}B_{1} A_{2}B_{2} ⋯ A_{m}B_{m} form a linear basis of this algebra. Consider the linear morphisms defined by setting

\underrightarrow{φ} : T → B_{n}[X], \underrightarrow{φ} (A_{1}B_{1} ⋯ A_{m}B_{m}) = \underrightarrow{A_{1}B_{1}} ⋯ \underrightarrow{A_{m}B_{m}}|_{n},
\underline{φ} : T → Super_{n}[X |X], \underline{φ} (A_{1}B_{1} ⋯ A_{m}B_{m}) = A_{1}B_{1} ⋯ \underline{A_{i}B_{i}} ⋯ A_{m}B_{m},
\underleftarrow{φ} : T → [X]_{n}B, \underleftarrow{φ} (A_{1}B_{1} ⋯ A_{m}B_{m}) = \underleftarrow{A_{1}B_{1}} ⋯ \underleftarrow{A_{m}B_{m}}|_{n};

we explicitly notice that the second map is well-defined, by Proposition 1. The crucial fact is that all these morphisms have the same kernel

Ker = Ker (\underrightarrow{φ}) = Ker (\underline{φ}) = Ker (\underleftarrow{φ}) ,

by Proposition 2; this kernel is a two-sided ideal of T, since \underrightarrow{φ} and \underleftarrow{φ} are algebra morphisms.

We consider the quotient algebra

\overline{T} = \frac{T}{Ker}.

In this algebra, we have the identity

A_{1}B_{1} A_{2}B_{2} ⋯ A_{m}B_{m} = A_{1}B′_{1} A′_{2}B′_{2} ⋯ A′_{m}B_{m},

under the assumption (B_{1}, A_{2}) ≃ (B′_{1}, A′_{2}), ..., (B_{m−1}, A_{m}) ≃ (B′_{m−1}, A′_{m}), and the common value of these expressions is zero unless B_{1} ≡ A_{2}, ..., B_{m−1} ≡ A_{m}. We have an expansion

A_{1}B_{1} A_{2}B_{2} ⋯ A_{m}B_{m} = ∑_{A′≡A} ζ_{A′} A′Q QB_{m},

where A′ ranges among the words equivalent to A_{1}, Q is a word with pairwise distinct positive entries which do not occur neither in A_{1} nor in B_{m}, and the coefficients ζ_{A′} are integers. Therefore, the algebra is linearly generated by the products AQ QB, with Q a word with pairwise distinct positive entries which do not occur neither in A nor in B.

In the algebra \overline{T}, for any pair (R, S) of tableaux of the same shape λ ⊢ n, we have three basic objects, given by

RC CS,  RC CD DS,  RD DS,

where C and D are any coDeruyts and Deruyts tableaux of shape λ; the images of these objects under the mappings \underrightarrow{φ}, \underline{φ}, \underleftarrow{φ} are the left Capelli rows, the bitableaux, the right Capelli rows, the left Young–Capelli symmetrizers, the symmetrized bitableaux, ...

The basic triangularity and nondegeneracy relations are

A_{1}B_{1} ⋯ A_{s}B_{s} = \{ \begin{matrix}0 & if sh(C) ≰ sh(D) \\ ϑ CD & if sh(C) = sh(D) \end{matrix} , (A_{1} ≡ C, B_{s} ≡ D)

CP QD = \{ \begin{matrix}0 & if P ≰ Q \\ ±c_{P}^{r}! c_{P}^{c}! CD & if P = Q \end{matrix} , (P, Q ∈ 𝔰𝔱)

CḊ ḊĊĊD = (−1)^{\binom{n}{2}}h_{λ} CD (sh(C) = … = sh(D) = λ).

There are also identities coming from the Straightening Laws (compare with [28], [10, 11]).

Finally, we notice that the group algebra ℚ[𝕊_{n}] of the symmetric group 𝕊_{n} occurs in a natural way in this context. Specifically, let P = x_{1}x_{2} ⋯ x_{n} be a word in pairwise distinct positive symbols; then the subspace \overline{S} linearly generated by the products

P_{1}Q QP_{2},

where P_{1} and P_{2} range through the words equivalent to P, and Q is a word in pairwise distinct positive symbols not occurring in P, is a subalgebra of the algebra \overline{T}.

The map Φ that sends the product P′Q QP′′, where

P′ = x_{i_{1}}x_{i_{2}} ⋯ x_{i_{n}}, P′′ = x_{j_{1}}x_{j_{2}} ⋯ x_{j_{n}},

to the the permutation

( \begin{matrix}i_{1} i_{2} ⋯ i_{n} \\ j_{1} j_{2} ⋯ j_{n} \end{matrix} )

induces an algebra isomorphism from the algebra \overline{S} to the group algebra ℚ[𝕊_{n}] of the symmetric group 𝕊_{n}. In this isomorphism, the objects of Young–Capelli type associated to pairs of multilinear tableaux in the symbols x_{1}, x_{2}, …, x_{n} are sent to the classical Young symmetrizers.

References

[1] A. Berele, A. Regev, Hook Young diagrams, combinatorics and representations of Lie superalgebras, Bull. Amer. Math. Soc. 8 (1983), 337–339.

[2] A. Berele, A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. Math. 64 (1987), 118–175.

[3] E. F. Beschler, D. A. Buchsbaum, J. T. Schwartz, R. P. Stanley, B. D. Taylor, M. Waterman, Gian-Carlo Rota (1932-1999), Notices Amer. Math. Soc. 47 (2000), no. 2, 203–216.

[4] H. Boerner, Representation of groups with special consideration for the needs of modern physics, Translated from the German by P. G. Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr. North–Holland Publishing Co., Amsterdam; Interscience Publishers, a division of John Wiley & Sons, Inc., New York, 1963.

[5] F. Bonetti, D. Senato, A. Venezia, The Robinson–Schensted correspondence for the fourfold algebra, Boll. Un. Mat. Ital. B (7) 2 (1988), no. 3, 541–554.

[6] A. Brini, Combinatorics, superalgebras, invariant theory and representation theory, Séminaire Lotharingien Combin. 55 (2007), Article B55g, 117 pp.

[7] A. Brini, R. Q. Huang, A. Teolis, The umbral symbolic method for supersymmetric tensors, Adv. Math. 96 (1992), no. 2, 123–193.

[8] A. Brini, A. Palareti, A. Teolis, Gordan–Capelli series in superalgebras, Proc. Natl. Acad. Sci. USA 85 (1988), 1330–1333.

[9] A. Brini, A. Teolis, Young—Capelli symmetrizers in superalgebras, Proc. Natl. Acad. Sci. USA 86 (1989), 775–778.

[10] A. Brini, A. Teolis, Capelli bitableaux and ℤ−forms of general linear Lie superalgebras, Proc. Natl. Acad. Sci. USA 87 (1990), 56–60.

[11] A. Brini, A. Teolis, Capelli’s theory, Koszul maps, and superalgebras, Proc. Natl. Acad. Sci. USA 90 (1993), 10245–10249.

[12] A. Brini, F. Regonati, A. Teolis, The method of virtual variables and representations of Lie superalgebras, in Clifford Algebras – Applications to Mathematics, Physics, and Engineering (Ablamowicz, R., ed.), Progress in Mathematical Physics, vol. 34, Birkhäuser, Boston, 2004, 245–263.

[13] A. Brini, F. Regonati, A. Teolis, Combinatorics and representations of general linear Lie superalgebras over letterplace superalgebras, in Computer Algebra and Clifford Algebra with Applications (H. Li, P. Olver, G. Sommer, Eds.), Lecture Notes in Computer Sciences, vol. 3519, Springer-Verlag, 2005, pp. 239–257.

[14] A. Brini, F. Regonati, A. Teolis, Combinatorics, transvectants and superalgebras. An elementary constructive approach to Hilbert’s finiteness theorem, Advances in Appl. Math. 37/3 (2006), 287–308; special issue dedicated to A.Regev (A. Berele and Y. Roichman, eds.).

[15] A. Capelli, Lezioni sulla teoria delle forme algebriche, Pellerano, Napoli, 1902.

[16] W. Chan, Invariant Theoretical Applications of Supersymmetric Algebra, Ph.D. thesis, M.I.T., 1995.

[17] S.-J. Cheng, W. Wang, Howe duality for Lie superalgebras, Compositio Math. 128 (2001), no 1, 55–94.

[18] M. Clausen, Letter place algebras and a characteristic-free approach to the representation theory of the general linear and symmetric groups I. Adv. Math. 33 (1979), 161–191.

[19] M. Clausen, Letter place algebras and a characteristic-free approach to the representation theory of the general linear and symmetric groups II. Adv. Math. 38 (1980), no. 2, 152–177.

[20] M. Clausen, Dominance orders, Capelli operators, and straightening of bideterminants. European J. Combin. 5 (1984), no. 3, 207–222.

[21] J. Désarménien, An algorithm for the Rota straightening formula, Discrete Math. 30 (1980), 51–68.

[22] J. Désarménien, J. P. S. Kung, G.-C. Rota, Invariant theory, Young bitableaux and Combinatorics, Adv. Math. 27 (1978), 63–92.

[23] P. Doubilet, G.-C. Rota, J. A. Stein, On the foundations of combinatorial theory IX. Combinatorial methods in invariant theory, Studies in Appl. Math. 53 (1974), 185–216.

[24] F. D. Grosshans, The symbolic method and representation theory. Adv. Math. 98 (1993), no. 1, 113–142.

[25] F. D. Grosshans, The work of Gian-Carlo Rota on invariant theory, Algebra Universalis 49 (2003), no. 3, 213–258.

[26] F. D. Grosshans, G.-C. Rota, J. A. Stein, Invariant theory and superalgebras, Amer. Math. Soc., Providence, RI, 1987.

[27] R. Q. Huang, N. L. White, Straightening coefficients in the supersymmetric letter-place algebra, J. Algebra 171 (1995), no. 3, 655–675.

[28] J.-L. Koszul, Les algèbres de Lie graduées de type 𝔰𝔩(n, 1) et l’opérateur de A. Capelli, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 2, 139–141.

[29] H. Kraft,  C. Procesi,  Classical Invariant Theory. A Primer,  1996;

    http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf.

[30] R. La Scala, V. Nardozza, D. Senato, Super RSK-algorithms and super plactic monoid, Internat. J. Algebra Comput. 16 (2006), no. 2, 377–396.

[31] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.

[32] C. Procesi, Lie groups. An approach through invariants and representations, Universitext. Springer, New York, 2007.

[33] D. E. Rutherford, Substitutional Analysis, Edinburgh University Publications: Science and Mathematics, Edinburgh-London, 1948.

[34] M. Scheunert, The Theory of Lie Superalgebras: An Introduction, Lecture Notes in Math., vol. 716, Springer Verlag, New York, 1979.

[35] A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras Gl(n, m) and Q(n), (Russian) Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430.

[36] A. N. Sergeev, An analog of the classical invariant theory for Lie superalgebras I. Michigan Math. J. 49 (2001), no. 1, 113–146.

[37] A. N. Sergeev, An analog of the classical invariant theory for Lie superalgebras II. Michigan Math. J. 49 (2001), no. 1, 113–146, 147–168.

[38] H. W. Turnbull, The Theory of Determinants, Matrices and Invariants, Blackie, London, 1950.

[39] A. H. Wallace, Invariant matrices and the Gordan–Capelli series, Proc. London Math. Soc. 2 (1952), 98–127.

[40] H. Weyl, The Classical Groups, Princeton Univ. Press, Princeton, NY, 1946.