Publ. I.R.M.A. Strasbourg, 1984, 229/S–08 Actes 8^{e} Sémininaire Lotharingien, p. 63–70

AN UMBRAL CALCULUS FOR POLYNOMIALS

CHARACTERIZING U(n) TENSOR PRODUCTS

BY

Stephen C. MILNE (^{∗})

This talk reports on work to appear in [1] that has been done jointly with L.C. BIEDENHARN and R.A. GUSTAFSON.

In this lecture we continue the study of the connection between the invariant polynomials

(1)  \begin{aligned}_{μ}G_{q}^{(n)} & ≡ _{μ}G_{q}^{(n)}((x_{ij} + Δ_{1})) \\ & ≡ _{μ}G_{q}^{(n)}(Δ_{1}, …, Δ_{n}; x_{12}, x_{23}, …, x_{n−1,n}, x_{n,1}). \end{aligned}

characterizing U(n) tensor operators ⟨p, q, …, q, 0, …, 0⟩, and the classical theory of symmetric functions as presented in [9], that was established in [5]. The polynomial _{μ}G_{q}^{(n)}(X) arise naturally in the application of symmetry groups to mathematical physics. One such problem, with applications to spectroscopy at all levels, is the construction of a suitable basis for the set of all bounded operators mapping the set of all unitary irreducible representation spaces of the group into itself. The precise problems that give rise to _{μ}G_{q}^{(n)}(X) are motivated in more detail and put into a broader mathematical setting in [2–4].

The above irrep label ⟨p, q, …, q, 0, …, 0⟩ consists of one p, μ q’s, and n − μ − 1 0’s. Furthermore, q determines p since Δ_{1} + ⋯ + Δ_{n} = p + μq and we are given q, Δ_{1} + ⋯ + Δ_{n} and μ.

Recently in [5–7] an alternate method for explicitly writing down μG_{q}^{(n)}(X) in polynomial form has been given. These methods provide a direct connection between the classical theory of symmetric functions as presented in [9], and the symmetries satisfied by a family of U(n) invariant polynomials even more general then _{μ}G_{q}^{(n)}(X). In [5] it is shown that after the change of variables

(2)  Δ_{i} = γ_{i} − δ_{i}  x_{i,i+1} = δ_{i} − δ_{i+1},

(^{∗}) Partially supported by a Sloan Foundation Fellowship and NSF grant MCS–8102032

S.C. MILNE

μG_{q}^{(n)}(, Δ_{i},; , x_{i,i+1},) becomes an integral linear combination of products of Schur functions S_{α}(, γ_{i},) · S_{β}(, δ_{i},) in the variables {γ_{1}, …, γ_{n}} and {δ_{1}, …, δ_{n}}, respectively. (See equation (6) for a definition of S_{λ}(x_{1}, …, x_{n}).) That is, it is directly proved that _{μ}G_{q}^{(n)}(, Δ_{i},; , x_{i,i+1},) is a bisymmetric polynomial in the variables {γ_{1}, …, γ_{n}} and {δ_{1}, …, δ_{n}} with integer coefficients. This motivated the study in [5] of the yet more general bisymmetric polynomials

(3)  _{μ}^{m}G_{q}^{(n)}(γ; δ) ≡ _{μ}^{m}G_{q}^{(n)}(γ_{1}, …, γ_{n}; δ_{1}, …, δ_{m})

which are a common generalization of (2.2b) of [2] and equation (2.17) of [10] with no numerator parameters. These polynomials are given by

Definition 4. — Given that _{μ}^{m}G_{0}^{(n)}(γ; δ) ≡ 1, we uniquely determine _{μ}^{m}G_{q}^{(n)}(γ; δ) by means of

(5) _{μ}^{m}G_{q}^{(n)}(, γ_{i},; , δ_{i},) = ∑_{{S⊂I_{n} \atop ∥S∥=μ+1}}(−1)^{μ+1+Σ(S)} ∏_{{i<j \atop i∈S, j∈S^{c}}} (γ_{i} − γ_{j})^{−1}

× ∏_{{i<j \atop i∈S^{c}, j∈S}} (γ_{i} − γ_{j})^{−1} ∏_{{i=1 \atop i∈S}}^{n} ∏_{l=1}^{m}(γ_{i} − δ_{l}) · _{μ}^{m}G_{q−1}^{(n)}(, γ_{i} − χ(i ∈ S),; , δ_{i},),

where S ⊂ I_{n} is a (μ + 1)-element subset of {1, 2, …, n}, Σ(S) denotes the sum of the elements in S, and χ(A) is 1 if statement A is true and 0, otherwise.

At this point we need to review some basic facts about the Schur functions S_{λ}. Let λ = (λ_{1}, λ_{2}, …, λ_{r}, …) be a partition, i.e., a (finite or infinite) sequence of nonnegative integers in decreasing order, λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{r} ⋯ such that only finitely many of the λ_{i} are nonzero. The number of nonzero λ_{i}, denoted by l(λ), is called the length of λ. If ∑ λ_{i} = n, then λ is called a partition of weight n, denoted by |λ| = n. Given a partition λ = (λ_{1}, …, λ_{n}) of length ≤ n, the Schur fonctions S_{λ} are defined by

(6)  S_{λ}(x_{1}, …, x_{n}) = \frac{det(x_{i}^{λ_{j}+n−j})_{1≤i,j≤n}}{det(x_{i}^{n−j})_{1≤i,j≤n}} .

The determinant in the numerator of (6) is divisible in Z[x_{1}, …, x_{n}] by each of the differences (x_{i}−x_{j}), 1 ≤ i < j ≤ n, and hence by their product, which is the Vandermonde determinant

(7a)  ∏_{1≤i<j≤n}(x_{i} − x_{j}) = det(x_{i}^{n−j})_{1≤i,j≤n}

≡ V_{n}(x_{1}, …, x_{n}). (7b)

UMBRAL CALCULUS

Thus, the quotient in (6) is a symmetric polynomial in x_{1}, …, x_{n} with coefficient in Z. For example, S_{(n)} = h_{n} and S_{(1^{n})} = e_{n} where h_{n} and e_{n} are, respectively, the homogeneous and elementary symmetric functions of x_{1}, …, x_{n}.

If Z = m_{1} + ⋯ + m_{n}, with m_{i} monomials, then

(8)  S_{λ}(Z) ≡ S_{λ}(m_{1} + ⋯ + m_{n}) ≡ S_{λ}(x_{1}, …, x_{n})|x_{i} = m_{i}.

On the other hand, S_{λ}(−(Z)) is defined symbolically by

(9)  S_{λ}(−(Z)) ≡ S_{λ}(−(m_{1} + ⋯ + m_{n})) ≡ (−1)^{|λ|}S_{λ′}(Z),

where |μ| = λ_{1}+⋯+λ_{n} is the sum of the parts of λ and λ′ is the conjugate partition to λ. That is, λ′ = (λ′_{1}, λ′_{2}, …, λ′_{(λ_{1})}), with λ′_{i} = ∥{j | λ_{j} ≥ i}∥. For example (5, 2, 1) is the conjugate partition of (3, 2, 1, 1, 1). Note that S_{λ}(−(m_{1} + ⋯ + m_{n})) is not equal to S_{λ}(x_{1}, …, x_{n})|_{x_{i}=−m_{i}}. The definitions given by (8) and (9) are implicit in [9 ; see Remark (3.10) of p. 26].

If ρ = (ρ_{1}, …, ρ_{k}), where ρ_{i} are integers, then denote by (X)^{ρ} the monomial x_{1}^{ρ_{1}} … x_{k}^{ρ_{k}} where X = {x_{1}, …, x_{k}}. Let the permutations w ∈ S_{k} act on subscripts. For example, w(X)^{ρ} = x_{w(1)}^{ρ_{1}} … x_{w(k)}^{ρ_{k}}. Finally, let δ_{k} be the partition (k − 1, k − 2, …, 0) and m^{l} the partition consisting of l parts equal to m.

Fix μ and assume that m − n ≡ ν is a constant. Denote the sets of variables {γ_{1}, …, γ_{μ+1}}, {γ_{μ+2}, …, γ_{n}}, {γ_{1}, …, γ_{n}}, and {δ_{1}, …, δ_{m}} by A, B, E and F, respectively. We then have from [5] the following fundamental

Theorem 10. — Let _{μ}^{m}G_{q}^{(n)}(E; F) be defined as in (5). We then have

(11) _{μ}^{m}G_{q}^{(n)}(E; F) = \frac{(−1)^{\binom{μ+1}{2}}}{V_{n}(E)} ∑_{{λ=(λ_{1},...,λ_{μ+1}) \atop λ_{1}≤m}} S_{λ}(−F)

· { ∑_{w∈S_{n}} ϵ(w) · w[(A)^{m^{μ+1}+δ_{μ+1}−(λ_{μ+1},...,λ_{1})}(B)^{δ_{n−(μ+1)}}

· _{μ}^{m}G_{q−1}^{(n)}(γ_{1} − 1, …, γ_{μ+1} − 1, B; F)]},

where, without loss of generalitly, w acts only on γ_{1}, …, γ_{μ+1} when applied to _{μ}^{m}G_{q−1}^{(n)}, and ϵ(w) is the sign of the permutation w.

Starting with (11) and making direct use of the new symmetries discovered in [5], it is shown in [1] that the bisymmetric polynomials _{μ}^{m}G_{q}^{(n)}(E; F) are an integral linear commbination of Schur functions S_{λ}(E − F) in the symbol E − F, where E − F denotes the difference of the two sets of variables E and F. Making use of properties of skew Schur functions S_{λ/μ} and S_{λ}(E − F) one puts together an umbral calculus for _{μ}^{m}G_{q}^{(n)}(E; F). That is, working entirely with polynomials, one uniquely determines _{μ}^{m}G_{q}^{(n)} from _{μ}^{m}G_{q−1}^{(n)} and combinatorial rules (such as the Littlewood-Richardson rule [9]) involving Ferrers diagrams (i.e. partitions). The deepest part of this umbral calculus is a summation theorem, involving many different aspects of the theory of Schur functions, which essentially reduces the double sum in (11) to the single term in which λ is the empty partition and w is the identity permutation. Here, we recall a general theorem form [1] which illustrates how the structure of _{μ}^{m}G_{q}^{(n)}(E; F) “stabilizes” as ν increases while m − n = ν remains fixed, state the umbral claculus for _{μ}^{m}G_{q}^{(n)}(E; F), and just give the final formulas for _{μ}^{m}G_{1}^{(n)}(E; F) and _{μ}^{m}G_{2}^{(n)}(E; F) which are a direct consequence of the umbral calculus and work in [5].

Before giving these formulas we need some more notation. Let λ, ρ be partitions. The skew Schur function S_{λ/ρ} is defined by

(12)  S_{λ/ρ}(E) = ∑_{ν⊂λ} c_{ρν}^{λ} · S_{ν}(E),

where ν ⊂ λ means ν_{i} ≤ λ_{i} for all i ≥ 1, and the integers c_{ρν}^{λ} are the Littlewood-Richardson rule coefficients determined by

(13)  S_{ρ}(E) · S_{ν}(E) = ∑_{ν} c_{ρν}^{λ}S_{λ}(E).

Note that S_{λ/0} = S_{λ}, where 0 denotes the zero partition. Also, c_{ρν}^{λ} = 0 unless |λ| = |ρ| + |ν|, so that S_{λ/ρ} is homogeneous of degree |λ| − |ρ|, and is zero if |λ| < |ρ|. In fact, S_{λ/ρ} = 0 unless ρ ⊂ λ.

Given the sets of variables E and F, let E + F be the set union {γ_{1}, …, γ_{n}, δ_{1}, …, δ_{m}}. There is then the following classical result

(14)  S_{λ}(E + F) = ∑_{ρ⊂λ} S_{λ/ρ}(E)S_{ρ}(F).

It is immediate from (9) and (14) that

(15a)  S_{λ}(E − F) = ∑_{ρ⊂λ} S_{ρ/λ}(E)S_{ρ}(−F)

= ∑_{ρ⊂λ}(−1)^{|ρ|}S_{ρ/λ}(E) · S_{ρ′}(F), (15b)

where ρ′ is the conjugate partition to ρ. The above classical formulas can be found in Chapter I of [9].

We now state a fundamental result of [1].

Theorem 16. — Fix μ ≥ 0 and k = m − n. There are integers a_{λ,q}, λ a partition, q a non-negative integer such that

(17)  _{μ}^{m}G_{q}^{(n)}(E; F) = ∑_{λ} a_{λ,q}S_{λ}(E − F),

where |λ| ≤ (μ + 1)(μ + 1 + k)q and λ ⊂ [(μ + 1 + k)q]^{(μ+1)q}.

If n ≥ (μ + 1)q and m ≥ (m + 1 + k)q, then a_{λ,q} is independent of n. More generally, if S_{λ}(E − F) ≠ 0, then a_{λ,q} is independent of n.

A similar “stabilization” theorem involving S_{α}(E) · S_{β}(F) instead of S_{λ}(E − F) was proved in [5]. Just as in [5] it follows from (17) that the n = (μ + 1)q and m = (μ + 1 + m − n)q case of (17) gives the correct formula for _{μ}^{m+l}G_{q}^{(n+l)}(E; F) for all integers l, when the sets E and F contain (n + l) and (m + l) variables, respectively.

Before stating the umbral calculus for _{μ}^{m}G_{q}^{(n)}(E; F) we need :

Definition 18. — Let α, β and γ be the three partitions α = (α_{1}, α_{2}, …, α_{n}), β = (β_{1}, β_{2}, …, β_{n}), γ = (γ_{1}, γ_{2}, …, γ_{n}). The sum α + β and direct sum α ⊕ γ are defined by

(19a)  α + β = (α_{1} + β_{1}, α_{2} + β_{2}, …, α_{n} + β_{n})

and

(19b)  α ⊕ γ = (α_{1}, α_{2}, …, α_{n}, γ_{1}, γ_{2}, …, γ_{m}).

Note that α + β is a partition while α ⊕ γ may not be. In general α ⊕ γ is just a (n + m)-tuple.

Definition 20. — Let ϕ ≡ (ϕ_{1}, ϕ_{2}, …, ϕ_{n}) be an n-tuple of nonnegative integers that is not necessarily a partition. If all the coordinates of (ϕ + δ_{n}) = (ϕ_{1} + n − 1, ϕ_{2} + n − 2, …, ϕ_{n−1} + 1, ϕ_{n}) are distinct, then there is a unique permutation σ_{ϕ} ∈ S_{n} that orders the parts of (ϕ + δ_{n}) in decreasing order. Denote the resulting partition (with distinct parts) by σ_{ϕ}(ϕ + δ_{n}). In addition, let (ϕ)_{σ_{ϕ}} be the partition

(21)  (ϕ)_{σ_{ϕ}} ≡ σ_{ϕ}(ϕ + δ_{n}) − δ_{n}.

Now, given any n-tuple ϕ = (ϕ_{1}, ϕ_{2}, …, ϕ_{n}), we determine S_{ϕ}(E − F) by means of

(22a)  S_{ϕ}(E − F), if the parts of (ϕ + δ_{n}) are not distinct.

Otherwise,

(22b)  S_{ϕ}(E − F) = ϵ(σ_{ϕ}) · S_{(ϕ)_{σϕ}}(E − F),

where (ϕ)_{σ_{ϕ}} is the partition given by (21) and ϵ(σ_{ϕ}) is the sign of the permutation σ_{ϕ}. (Note that if ϕ is a partition, then the parts of (ϕ + δ_{n}) are distinct, δ_{ϕ} is the identity permutation, and the right-side of (22b) is simply (S_{ϕ}(E − F).)

Theorem 23. — One has

(24)  S_{λ}(X + Y) = ∑_{π, α⊂λ} c_{πα}^{λ}S_{α}(Y)S_{π}(X),

where c_{πα}^{λ} are the L–R coefficients in (13), and X and Y are two (signed) sums of monomials.

Theorem 25 (A. Lascoux). — One has

(26)  S_{π}(−1 + γ_{1}, …, −1 + γ_{μ+1}) = ∑_{ν⊂π} d_{πν}S_{ν}(A),

where

(27)  d_{πν} = (−1)^{|π|−|ν|} det [\binom{π_{i} + μ + 1 − i}{ν_{j} + μ + 1 − j}]_{1≤i, j≤μ+1} ,

and

A = {γ_{1}, γ_{2}, …, γ_{μ+1}}.

THEOREM 23 appears in [9 ; see eq. (5.9) on p. 41] and THEOREM 25 is due to A. LASCOUX [8] and can be found in [9 ; see ex. 10, p. 30].

The first step of the umbral calculus is to compute

(28)  _{μ}^{m}G_{q−1}^{(n)}(γ_{1} − 1, γ_{2} − 1, …, γ_{μ+1} − 1, B; F).

This is accomplished by means of the q − 1 case of THEOREM 16, and THEOREMS 23 and 25. The second (and deepest) part of the umbral calculus is to replace S_{ν}(A) · S_{α}(B − F) in the sum giving (28) by

(29)  S_{(((μ+1+m−n)^{μ+1}+ν)⊕α)}(E − F).

where (29) is determined by the ϕ = (((μ + 1 + m − n)^{μ+1} + ν) ⊕ α) case of Definition 20. The resulting sum is equal to _{μ}^{m}G_{q}^{(n)}(E; F). That is, we have

Theorem 30 (umbral calculus). — Let _{μ}^{m}G_{q−1}^{(n)}(E; F) be given by the q − 1 case of (17). For technical reasons assume that the following inequalities hold : (μ + 1)q ≤ n, m ≥ μ + 1, m − n = k is constant, l(ν) ≤ μ + 1, and l(α) ≤ n − (μ + 1). We then have

(31) _{μ}^{m}G_{q}^{(n)}(E; F) = (−1)^{\binom{μ+1}{2}} ∑_{λ} a_{λ,q−1}

· \{ ∑_{{π, α⊂λ \atop ν⊂π}} c_{πα}^{λ} · d_{πν} · S_{(((μ+1+m−n)^{μ+1}+ν)⊕α)}(E − F)\},

where |λ| ≤ (μ + 1)(μ + 1 + k)(q − 1) and λ ⊂ [(μ + 1 + k)(q − 1)]^{(μ+1)(q−1)} and where S_{(((μ+1+m−n)^{μ+1}+ν)⊕α)}(E − F) is determined by the ϕ =

(((μ + 1 + m − n)^{μ+1} + ν) ⊕ α) case of Definition 20, c_{πα}^{λ} are the L–R coefficients in (13), and the d_{πν} are defined by (27).

THEOREM 1.22 of [5] expressed _{μ}^{m}G_{1}^{(n)}(E; F) as a sum of products of Schur functions S_{α}(F)·S_{β}(F). Using (15b) to rewrite this sum immediately gives

(32)  _{μ}^{m}G_{1}^{(n)}(E; F) = (−1)^{\binom{μ+1}{2}}S_{(μ+1+m−n)^{μ+1}}(E − F),

where (μ + 1 + m − n)^{μ+1} denotes the partition consisting of (μ + 1) parts equal to (μ + 1 + m − n).

In [5] it took three pages to write _{1}^{2}G_{2}^{(n)}(E; F) as a sum of products of Schur functions. Starting with (32) and making use of THEOREM 30 it was shown in [1] that

(33) _{1}^{2}G_{2}^{(n)}(E; F) = {S_{2^{2}}(D)}^{2} + {−[2S_{4,3}(D) + 2S_{2^{3},1}(D)

+ 3S_{3,2^{2}}(D) + 3S_{3^{2},1}(D) + S_{4,3,1}(D) + S_{3,2,1^{2}}(D)] + [S_{4,2}(D) + S_{2^{2},1^{2}}(D) + 3S_{2^{3}}(D) + 3S_{3^{2}}(D) + 3S_{3,2,1}(D)]

+ [2S_{3,2}(D) + 2S_{2^{2},1}(D)] + S_{2^{2}}(D)},

where D denotes the set difference E − F.

Further applications of THEOREMS 16 and 30, as well as detailed proofs, can be found in [1].

REFERENCES

[1] BIEDENHARN (L.C.), GUSTAFSON (R.A.) and MILNE (S.C.). — An umbral calculus for polynomials characterizing U(n) tensor operators, Advances in Math., t. 51, 1984, p. 36–90.

[2] BIEDENHARN (L.C.), HOLMAN III (W.) and MILNE (S.C.). — The invariant polynomials characterizing U(n)tensoroperators⟨p, q, …, q, 0, …, 0⟩ having maximal null space, Advances in Appl. Math., t. 1, 1980, p. 390–472.

[3] BIEDENHARN (L.C.) and LOUCK (J.D.). — Angular Momentum in Quantum Physics : Theory and Applications. — Reading, Addison-Wesley (Encyclopedia of Math. Appl., 8), 1981.

[4] BIEDENHARN (L.C.) and LOUCK (J.D.). — The Racah-Wigner Algebra in Quantum Theory. — Reading, Addison-Wesley (Encyclopedia of Math. Appl., 9), 1981.

[5] GUSTAFSON (R.A.) and MILNE (S.C.). — Schur functions and the invariant characterizing U(n) tensor operators, Advances in Appl. Math., t. 4, 1983, p. 422–478.

[6] GUSTAFSON (R.A.) and MILNE (S.C.). — Schur functions, Good’s identity, and hypergeometric series well-poised in SU(n) tensor operators, Advances in Math., t. 48, 1983, p. 177–188.

[7] GUSTAFSON (R.A.) and MILNE (S.C.). — A new symmetry for Biedenharn’s G-functions and classical hypergeometric series, Advances in Math., in press.

[8] LASCOUX (A.). — Classes de Chern d’un produit tensoriel, C.R. Acad. Sci. Paris, t. 268A, 1978, p. 385–387.

[9] MACDONALD (I.G.). — Symmetric Functions and Hall Polynomials. — Oxford Univ. Press, 1979.

[10] MILNE (S.). — Hypergeometric series well-poised in SU(n) and a generalization of Biedenharn’s G-functions, Advances in Math., t. 36, 1980, p. 169–211.

Stephen C. Milne,

Department of Mathematics, Texan A & M University, College Station, Texas 77843, U.S.A.

now at :

Department of Mathematics,

Ohio State University Columbus, Ohio, U.S.A.