Séminaire Lotharingien de Combinatoire 65 (2012), Article B65f

Doubly-refined enumeration of Alternating Sign Matrices

and determinants of 2-staircase Schur functions

Philippe Biane, Luigi Cantini, and Andrea Sportiello

Abstract. We prove a determinantal identity concerning Schur functions for 2staircase diagrams λ = (ℓn+ℓ′, ℓn, ℓ(n−1)+ℓ′, ℓ(n−1), …, ℓ+ℓ′, ℓ, ℓ′, 0). When ℓ = 1 and ℓ′ = 0 these functions are related to the partition function of the 6-vertex model at the combinatorial point and hence to enumerations of Alternating Sign Matrices. A consequence of our result is an identity concerning the doubly-refined numbers of Alternating Sign Matrices.

1. Introduction

1.1. Alternating Sign Matrices. An alternating sign matrix (ASM) is a square matrix with entries in {−1, 0, +1}, such that along each row and along each column, if one forgets the 0’s, the +1’s and −1’s alternate, and the sum of the entries of each row and of each column is equal to 1. It is a famous combinatorial result that the number of such matrices of size n is

(1.1)  A_{n} = ∏_{j=0}^{n−1} \frac{(3j + 1)!}{(n + j)!} = 1, 2, 7, 42, 429, …

After having been a conjecture for several years [12], this was first proven by Zeilberger in [17], and a simpler proof was given by Kuperberg [9], using a connection with the 6-Vertex Model of statistical mechanics, and an appropriate multivariate extension of the mere counting function A_{n}. A vivid account can be found in [1].

It follows easily from the definition that an alternating sign matrix has exactly one +1 in its first (and last) row (and column). Thus we have a sensible four-variable refined statistics, for these four positions in {1, …, n}^{4}, together with their projections on a smaller number of variables. After the dihedral symmetry of the square is taken into account, what remains is a single one-variable statistics (exhibiting a round formula), and two doubly-refined statistics: one, A_{ij}^{n}, for the first and last row (or the rotated case), and one, B_{ij}^{n}, for the first row and column (or the three rotated cases), see Figure 1, left. Matrices A^{n} of these doubly-refined alternating sign matrix numbers A_{ij}^{n}

2010 Mathematics Subject Classification. Primary 05E05; Secondary 05A15, 15A15, 82B23.

Key words and phrases. Alternating sign matrices, Schur functions, compound determinants.

k  j i

Figure 1. Left: a typical alternating sign matrix of size n = 10 (empty cells, disks and diamonds stand for 0, +1 and −1 entries, respectively). This matrix contributes to the statistics A_{ij}^{n} and B_{ik}^{n}, with (i, j, k) = (6, 4, 5). Right: empty cells are replaced by scale-shaped tiles, as to produce a valid tiling (i.e., concavities of neighbouring arcs do match). The direction of the tip specifies if the cell is of type NW, NE, SE or SW.

for n = 1, 2, 3, 4, 5 are given by

A^{1} = (1);  A^{2} = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix});  A^{3} = (\begin{matrix}0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 0 \end{matrix});

A^{4} = (\begin{matrix}0 & 2 & 3 & 2 \\ 2 & 4 & 5 & 3 \\ 3 & 5 & 4 & 2 \\ 2 & 3 & 2 & 0 \end{matrix});  A^{5} = (\begin{matrix}0 & 7 & 14 & 14 & 7 \\ 7 & 21 & 33 & 30 & 14 \\ 14 & 33 & 41 & 33 & 14 \\ 14 & 30 & 33 & 21 & 7 \\ 7 & 14 & 14 & 7 & 0 \end{matrix} ) .

Of course, by definition ∑_{i,j} A_{ij}^{n} = A_{n}, i.e., 1, 2, 7, 42, 429, … for the cases above. A simple argument (if an alternating sign matrix contains a +1 in a corner, then the rows and columns not containing that +1 form an alternating sign matrix of size one less) implies that the sum of the entries along the first (and last) row (and column) gives A_{n−1}, i.e., 1, 1, 2, 7, 42, …, and that the bottom-left and top-right entries are A_{n−2}, i.e., 1, 1, 1, 2, 7, … These simple identities are linear. There exist also quadratic relations, of Plücker nature, relating these doubly-refined enumerations to A_{n} and the (singly-) refined enumerations (see, e.g., [16, 2]).

Let us now evaluate the determinant of these matrices:

det(A^{2}) = −1 = −1^{−1},  det(A^{3}) = 1 = 2^{0},
det(A^{4}) = −7 = −7^{1},  det(A^{5}) = 1764 = 42^{2}, …

These small numerics suggest a relation that we prove in this paper.

Theorem 1.

(1.2)  det(A^{n}) = (−A_{n−1})^{n−3} .

This relation is non-linear. Its degree is neither fixed, nor bounded. What is fixed is what we could call “co-degree”, namely the system size, minus the degree (in analogy to the definition of co-dimension of a subspace). Relations of this nature seem to be a novelty for the subject at hand.

Our proof of the theorem above will result as corollary of a much more general result on certain Schur functions. To see why these two topics are connected, we have to recall Kuperberg’s solution of the Alternating Sign Matrix conjecture.

1.2. ASM, the 6-Vertex Model, and Schur functions. It follows from the connection with the 6-Vertex Model, that the generating function for a certain weighted enumeration of alternating sign matrices is given by a closed determinantal formula. Given an ASM B = {B_{ij}}_{1≤i,j≤n} and i, j such that B_{ij} = 0, we say that (i, j) is a north-west (NW) site (respectively NE, SE, SW) if, forgetting the zeroes, the next +1 entry along the same column is in north direction, and along the same row is in west direction (and analogously for the other three cases) — see the right part of Figure 1. Consider some complex-valued function μ_{n}(B) defined on n × n ASMs, and call

Z_{n} = ∑_{B} μ_{n}(B)

the corresponding generating function (in statistical mechanics μ_{n}(B) is a generalized Gibbs measure — an ordinary measure if it is real-positive and normalized — and Z_{n} is the partition function).

When μ_{n}(B) has the following factorized form, parametrized by 2n + 1 variables (x_{1}, …, x_{n}, y_{1}, …, y_{n}, q) = (x⃗, y⃗, q),

(1.3a)  μ_{n}(B; x⃗, y⃗, q) = ∏_{1≤i,j≤n} w_{i,j}(B) ;

(1.3b)  w_{i,j}(B) = \{ \begin{matrix}(q − q^{−1})\sqrt{x_{i}y_{j}} & B_{ij} = ±1; \\ q^{−1}x_{i} − qy_{j} & B_{ij} = 0, (i, j) is NW or SE; \\ −x_{i} + y_{j} & B_{ij} = 0, (i, j) is NE or SW; \end{matrix}

integrability methods, and a recursion due to Korepin [7], allowed Izergin [6] to establish a determinantal expression for the generating function Z_{n}(x⃗, y⃗, q) = ∑_{B} μ_{n}(B; x⃗, y⃗, q). In particular, this function is symmetric under 𝔖_{n} × 𝔖_{n} acting on row- and columnparameters x_{i} and y_{j}.

The evaluation of A_{n} is recovered if we set q = exp(\frac{2πi}{3}), x_{i} = q^{−1} for all i and y_{j} = q for all j, as in this case the local weights w_{i,j} become all equal to \sqrt{−3}, independently of B, and thus μ_{n}(B) becomes constant (i.e., the uniform measure, up to an overall factor).

Later on, it has been discovered [16, 14] that the value q = exp(\frac{2πi}{3}) (sometimes called the combinatorial point) has a special combinatorial property: Z_{n}(x⃗, y⃗, q) becomes fully symmetric under 𝔖_{2n} (acting on the 2n-tuple of qx_{i}’s and q^{−1}y_{j}’s together); more precisely, it is proportional to the Schur function associated to the Young diagram λ_{n} = (n − 1, n − 1, n − 2, n − 2, …, 1, 1, 0, 0) (see Appendix A for the definition), evaluated at {qx_{1}, …, qx_{n}, q^{−1}y_{1}, …, q^{−1}y_{n}} (see Figure 2, left, for a picture of this Young diagram). One consequence is that we have

(1.4)  A_{n} = 3^{−\binom{n}{2}}s_{λ_{n}}(1, 1, …, 1),

2n{  _{ℓ′} _{ℓ}

Figure 2. Left: the Young diagram λ_{n}, for n = 5. Right: the Young diagram λ_{n,ℓ,ℓ′}, for n = 5, ℓ = 3 and ℓ′ = 2.

and also the refined enumerations introduced above are related to specializations of this Schur function, in which some parameters are left as indeterminates.

In particular for the A_{ij}^{n}’s, defining the generating function

A_{n}(u, v) = ∑_{1≤i,j≤n} A_{ij}^{n} u^{i−1}v^{n−j} ,

one finds

(1.5)  A_{n}(u, v) = 3^{−\binom{n}{2}}(q^{2}(q + u)(q + v))^{n−1}s_{λ_{n}}(\frac{1+qu}{q+u}, \frac{1+qv}{q+v}, 1, …, 1) .

(The rational function \frac{1+qu}{q+u} originates from the ratio of w_{ij}(B) in the two last cases of (1.3b).)

A detailed analysis of the doubly-refined enumeration formula (1.5) restated in terms of multiple contour integrals, and the proof of a relation with a doubly-refined enumeration formula for totally-symmetric self-complementary plane partitions in a cube of size 2n, can be found in [5].

1.3. On the determinants of Schur functions. In this section we state a theorem concerning the determinant of a matrix whose entries are Schur functions s_{λ_{n}}. Not surprisingly, as these functions are related to ASM enumerations, e.g., through equations (1.4) and (1.5), this property will show up to be the structure behind Theorem 1, and conceivably, it has an interest by itself. For this reason, in this paper we pursue the task of stating and proving a much wider version of the aforementioned property, than the one that would suffice for Theorem 1. This leads us to introduce a wider family of Young diagrams.

We define the 2-staircase diagram λ_{n,ℓ,ℓ′}, for n ≥ 1, 0 ≤ ℓ′ ≤ ℓ, as

λ_{n,ℓ,ℓ′} = ((n − 1)ℓ + ℓ′, (n − 1)ℓ, (n − 2)ℓ + ℓ′, (n − 2)ℓ, …, ℓ′, 0);

i.e., (λ_{n,ℓ,ℓ′})_{2j−1} = (n − j)ℓ + ℓ′ and (λ_{n,ℓ,ℓ′})_{2j} = (n − j)ℓ (see Figure 2, right). We call the associated Schur polynomial, s_{λ_{n,ℓ,ℓ′}}(z_{1}, …, z_{2n}), a 2-staircase Schur function.

The name comes from the fact that this family of diagrams generalizes the wellknown family of staircase diagrams μ_{n,ℓ}

(1.6)  μ_{n,ℓ} = ((n − 1)ℓ, (n − 2)ℓ, …, ℓ, 0) .

The Schur functions s_{λ_{n}} are thus particular cases of 2-staircase Schur functions, corresponding to ℓ = 1 and ℓ′ = 0.

The polynomials s_{λ_{n,ℓ,ℓ′}} have been considered recently by Alain Lascoux. In particular, in [11, Lemma 13] they are shown to coincide with the specialization at q = exp(\frac{2πi}{ℓ+2}) of a certain natural extension of Gaudin functions.

In an apparently unrelated context we see the appearance of the polynomials s_{λ_{n,ℓ,ℓ′}}, for ℓ′ = 0 only. This context, analysed by Paul Zinn-Justin in [18], is the study of the solution of the qKZ equation related to the spin ℓ/2 representation of the quantum affine algebra U_{q}({ \atop 𝔰𝔩(2)}) with q = exp(\frac{2πi}{ℓ+2}). It is shown that, by taking the scalar product of the solution of the qKZ equation with a natural reference state, one obtains s_{λ_{n,ℓ,0}}.

As anticipated, our Theorem 1 will be a corollary of the following result, of independent interest, which exhibits a remarkable factorization of a determinant of 2-staircase Schur functions:

Theorem 2. Let N = ℓ(n − 1) + ℓ′ + 1. Let {x_{i}, y_{i}}_{1≤i≤N} be indeterminates, let f(z⃗, w_{1}, w_{2}) stand for f(z_{1}, …, z_{2n−2}, w_{1}, w_{2}), and, for an ordered N-tuple x⃗ = (x_{1}, x_{2}, …, x_{N}), let Δ(x⃗) = ∏_{i<j}(x_{i} − x_{j}) denote the usual Vandermonde determinant. Then

(1.7) det (s_{λ_{n,ℓ,ℓ′}}(z⃗, x_{i}, y_{j}))_{1≤i,j≤N}

= c(n, ℓ, ℓ′) Δ(x⃗)Δ(y⃗) ( ∏_{i=1}^{2n−2} z_{i}^{ℓ′(ℓ+1)}) s_{μ_{2n−2,ℓ+1}}^{ℓ}(z⃗) s_{λ_{n−1,ℓ,ℓ′}}^{ℓ(n−2)+ℓ′−1}(z⃗) .

The quantity c(n, ℓ, ℓ′) has its values in {0, ±1}. More precisely,

(1.8) c(n, ℓ, ℓ′) = \{ \begin{matrix}(−1)^{(n−1)\binom{ℓ+1}{2}+\binom{ℓ′+1}{2}} & if n = 1 or gcd(ℓ + 2, ℓ′ + 1) = 1, \\ 0 & if n > 1 and gcd(ℓ + 2, ℓ′ + 1) ≠ 1. \end{matrix}

Observe that, as is well known, the staircase Schur function s_{μ_{2n−2,ℓ+1}} can be further factorized. Let us recall the definition of the (bivariate homogeneous) Chebyshev polynomials (of the second kind)

(1.9)  U_{h}(x, y) = \frac{x^{h+1} − y^{h+1}}{x − y} = x^{h} + x^{h−1}y + ⋯ + y^{h} = ∏_{i=1}^{h}(x − ζ^{i}y) ,

where ζ is a primitive (h + 1)-st root of unity. One can write (cf. equation (A.6))

(1.10)  s_{μ_{N,h}}(z⃗) = ∏_{1≤i<j≤N} U_{h}(z_{i}, z_{j}) .

As Schur functions have several determinant representations (see Appendix A), the left-hand side quantity of the theorem is a “determinant of determinants”, a structure in linear algebra that is sometimes called a compound determinant [13, Ch. VI]. As we will see, the theory of compound determinants will play a crucial role in our proof.

Results of the form of the one in Theorem 2, or at least approaches to quantities as in the left-hand side of equation (1.7), already exist in the literature, although mostly with partitions of comparatively simpler structure. Cf. [11], where also a general approach is outlined. In particular, equations (23) and (24) in [11] have a form of striking similarity with our theorem above, while involving a rectangular partition r^{p} ≡ (r, r, …, r) (r is repeated p times), and the basic 1-staircase partition (r, r − 1, r − 2, …, 1, 0), respectively, and the unnumbered third equation after Corollary 9 of [11] (for which, however, no factorization is stated) has a similar structure to what will be the matrix of our analysis, with the only difference that it presents a Chebyshev polynomial in the denominator instead of in the numerator.

Theorem 2 is easily seen to hold for n = 1 and any (ℓ, ℓ′). This could seem a good base for an induction. However we use inductive arguments only for the minor task of determining the overall constant c(n, ℓ, ℓ′) in Section 4.2. On the other hand, in Section 4.1 we prove divisibility results, by a method reminiscent of the “exhaustion of factors” method described in Krattenthaler’s survey [8].

Note however that the factors s_{λ_{n−1,ℓ,ℓ′}} are polynomials of ‘large’ degree, ℓn(n − 1) + ℓ′n, with no factorizations as long as gcd(ℓ + 2, ℓ′ + 1) = 1 (we give a partial proof of this statement in Proposition 5 below — a full proof is not hard to achieve). Thus, in a sense, the tools we develop in Section 3 should be regarded as an extension of the exhaustion of factor method to the case in which we have an infinite family of determinantal identities, and some of the factors have an unbounded degree, scaling with the size parameter associated to the family.

Finally, let us add a few words on notations: throughout the paper, if z⃗ is a vector of length n (the length will be clear by the context), we write f(z⃗) as a shortcut for f(z_{1}, …, z_{n}), and f(z⃗, w_{1}, w_{2}, …) as a shortcut for f(z_{1}, …, z_{n}, w_{1}, w_{2}, …). We also write f(z⃗_{\i_{1}···i_{k}}, w_{1}, w_{2}, …) if the variables z_{i_{1}}, …, z_{i_{k}} are dropped from the list (z_{1}, …, z_{n}).

The paper is organized as follows. In Section 2 we show how to derive Theorem 1 from Theorem 2 specialized to ℓ = 1 and ℓ′ = 0. In Section 3 we present some preparatory lemmas for the proof of Theorem 2, which is presented in Section 4. Appendix A collects some basic definitions and facts on Schur functions, while in Appendix B we introduce an even larger class of staircase Schur functions, and study some of their properties.

2. Derivation of Theorem 1 from Theorem 2

For a polynomial f(x, y), denote by [x^{i}y^{j}]f(x, y) the coefficient of the monomial x^{i}y^{j}. We first state a simple but useful lemma.

Lemma 1. Let P (u, v) be a polynomial in two indeterminates, of degree at most n − 1 in each variable. Set P = ([u^{i−1}v^{j−1}]P (u, v))_{1≤i,j≤n}. Furthermore, let u_{i}, v_{j} be indeterminates. Then

(2.1)  det (P (u_{i}, v_{j}))_{1≤i,j≤n} = Δ(u⃗)Δ(v⃗) det P .

Proof. Let V (u⃗) denote the Vandermonde matrix (u_{j}^{i−1})_{1≤i,j≤n}. Then

det V (u⃗) = (−1)^{n(n−1)/2}Δ(u⃗),

and the matrix (P (u_{i}, v_{j}))_{1≤i,j≤n} is the product V (u⃗)^{T}P V (v⃗). □

This lemma implies that our Theorem 2 is equivalent to

(2.2) det ([x^{i}y^{j}]s_{λ_{n,ℓ,ℓ′}}(z⃗, x, y))_{0≤i,j≤ℓ(n−1)+ℓ′}

= c(n, ℓ, ℓ′) ( ∏_{i=1}^{2n−2} z_{i}^{ℓ′(ℓ+1)}) s_{μ_{2n−2,ℓ+1}}^{ℓ}(z⃗) s_{λ_{n−1,ℓ,ℓ′}}^{ℓ(n−2)+ℓ′−1}(z⃗)

(of course, with c(n, ℓ, ℓ′) as in (1.8)).

Now we proceed to the proof of Theorem 1. With u⃗ = (u_{1}, …, u_{n}), we compute

(2.3)  Δ(\{\frac{1+qu_{i}}{q+u_{i}} \}) = Δ(u⃗) (q^{2} − 1)^{\binom{n}{2}} ∏_{i}(q + u_{i})^{−(n−1)}.

It follows from Lemma 1, and equation (1.5), that

Δ(u⃗)Δ(v⃗) det(A_{ij}^{n}) = (2.4)

= (−1)^{\binom{n}{2}} det (\frac{(q^{2}(q + u_{i})(q + v_{j}))^{n−1}}{3^{\binom{n}{2}}} s_{λ_{n}}(\frac{1+qu_{i}}{q+u_{i}} , \frac{1+qv_{j}}{q+v_{j}} , 1, …, 1))_{1≤i,j≤n}

= (\frac{−q^{4}}{3^{n}} )^{\binom{n}{2}} ∏_{i=1}^{n} ((q + u_{i})(q + v_{i}))^{n−1} det (s_{λ_{n}}(\frac{1+qu_{i}}{q+u_{i}} , \frac{1+qv_{j}}{q+v_{j}} , 1, …, 1))_{1≤i,j≤n} .

Applying Theorem 2 with ℓ = 1, ℓ′ = 0, x_{i} = \frac{1+qu_{i}}{q+u_{i}}, and y_{j} = \frac{1+qv_{j}}{q+v_{j}} to the determinant on the right-hand side, and then (2.3), we obtain

(2.5) Δ(u⃗)Δ(v⃗) det(A_{ij}^{n}) = Δ(u⃗)Δ(v⃗) (−1)^{n−1+\binom{n}{2}} (\frac{(q − q^{2})^{2}}{3^{n}} )^{\binom{n}{2}}

× s_{μ_{2n−2,2}}(1, 1, …, 1) s_{λ_{n−1}}^{n−3}(1, 1, …, 1)

It should be noted that (q − q^{2})^{2} = −3. By the explicit evaluation of a staircase Schur function, equation (1.10), we have

(2.6)  s_{μ_{2n−2,2}}(1, 1, …, 1) = 3^{\binom{2n−2}{2}}.

Theorem 1 follows from (1.4), (2.5), and (2.6). □

3. Preliminary results

3.1. On the minor expansion of a sum of matrices. Consider k n×n matrices M^{(a)} = (M_{ij}^{(a)})_{1≤i,j≤n}, 1 ≤ a ≤ k, where the M_{ij}^{(a)} are indeterminates. Given an n × n matrix A and subsets I, J of [n] := {1, 2, …, n}, denote by A_{I,J} the restriction of A to rows in I and columns in J. Denote by I = (I_{1}, …, I_{k}) an ordered k-tuple of subsets I_{a} ⊆ [n] (possibly empty), forming a partition of [n]. For two such k-tuples I and J, say that they are compatible if |I_{a}| = |J_{a}| for all a ∈ {1, …, k}, and write I ∼ J in this case. Denote by ϵ(I, J) the sign of the permutation that reorders (I_{1}, …, I_{k}) into (J_{1}, …, J_{k}), with elements within the blocks in order. Then we have the following fact.

Proposition 1 (Minor expansion of a sum of matrices).

(3.1)  det ( ∑_{a=1}^{k} M^{(a)}) = ∑_{\begin{matrix}I,J \\ I∼J \end{matrix}} ϵ(I, J) ∏_{a=1}^{k} det M_{I_{a},J_{a}}^{(a)} .

Proof. Consider the full expansion of the determinant

det ( ∑_{a=1}^{k} M^{(a)}) = ∑_{σ∈𝔖_{n}} ϵ(σ) ∏_{i=1}^{n} ( ∑_{a=1}^{k} M_{i σ(i)}^{(a)})

= ∑_{σ∈𝔖_{n}} ∑_{b∈[k]^{n}} ϵ(σ) ∏_{i=1}^{n} M_{i σ(i)}^{(b(i))}.

To each pair (σ, b) in the linear combination above, we associate a pair (I, J) of compatible partitions by

(3.2)  I_{a} = {i : b(i) = a};  J_{a} = {j : b(σ^{−1}(j)) = a} .

So I is determined by b alone, and all the permutations σ producing the same J can be written as the “canonical” permutation τ that reorders (I_{1}, …, I_{k}) into (J_{1}, …, J_{k}) with elements within the blocks in order, acting from the left on a permutation ρ = ∏_{a} ρ_{a} ∈ 𝔖_{I_{1}} × ⋯ × 𝔖_{I_{k}}. The sign factorizes, ϵ(σ) = ϵ(τ) ∏_{a} ϵ(ρ_{a}), and ϵ(τ) = ϵ(I, J) by definition, thus

det ( ∑_{a=1}^{k} M^{(a)}) = ∑_{\begin{matrix}I,J \\ I∼J \end{matrix}} ϵ(I, J) ∏_{a} ∑_{ρ_{a}∈𝔖_{Ia}} ϵ(ρ_{a}) ∏_{i∈I_{a}} M_{i τ∘ρ_{a}(i)}^{(a)}.

For each index a, the sum over the permutations ρ_{a} produces the determinant of the appropriate minor. □

3.2. Bazin–Reiss–Picquet Theorem. In this section we recall the Bazin–Reiss– Picquet Theorem [13, pp. 193–195, § 202–204]. Take a triple (m, n, p) of integers m ≥ n ≥ p ≥ 0. Let S_{n,p} be the set of subsets of [n] of cardinality p. (Thus, |S_{n,p}| = \binom{n}{p}.) For a set I ∈ S_{n,p}, write I = {i_{1}, …, i_{p}} for the ordered list of elements.

Consider the m × n matrices A = (A_{ij})_{1≤i≤m, 1≤j≤n} and B = (B_{ij})_{1≤i≤m, 1≤j≤n}, and the m × (m − n) matrix C = (C_{ij})_{1≤i≤m, 1≤j≤m−n}, where the A_{ij}, B_{ij}, and C_{ij} are indeterminates. Write (X|Y) for the matrix resulting from taking all the columns of X, followed by all the columns of Y.

For a pair (I, J) ∈ S_{n,p} × S_{n,p}, define M^{I,J} as the matrix

M_{h,k}^{I,J} = \begin{cases}A_{h,k} & if k ≤ n, k ∉ I; \\ B_{h,j_{ℓ}} & if k = i_{ℓ}; \\ C_{h,k−n} & if n < k ≤ m; \end{cases}

(that is, replace the columns I of (A|C) by the columns J of B, in order). Define D_{I,J} = det M^{I,J}. Choose a total ordering of S_{n,p}, and construct the matrix D = (D_{I,J})_{I,J∈S_{n,p}}, which is of size \binom{n}{p}. Then the compound determinant det D does not depend on the chosen ordering, and it has the following factorization property.

Theorem 3 (Bazin–Reiss–Picquet). We have

(3.3)  det D = det(A|C)^{\binom{n−1}{p}} det(B|C)^{\binom{n−1}{p−1}} .

3.3. A divisibility corollary. A corollary of the Bazin–Reiss–Picquet Theorem is a divisibility result for a special family of determinants. Take m ≥ n ≥ k ≥ 0. Consider m indeterminates z_{i}, n indeterminates y_{j}, and 2nk indeterminates u_{i}^{a}, v_{i}^{a}, with 1 ≤ i ≤ n and 1 ≤ a ≤ k (u_{i}^{a}, v_{i}^{a} may possibly be elements in the polynomial ring in z_{i}’s and y_{j}’s). Take m polynomial functions f_{j}(x), and introduce the associated Slater determinant, that is, the antisymmetric polynomial

P (x⃗) = P (x_{1}, …, x_{m}) = det (f_{j}(x_{i}))_{1≤i,j≤m} .

A typical example is a shifted Vandermonde determinant,

P (x_{1}, …, x_{m}) = Δ_{λ}(x_{1}, …, x_{m}),

for λ a partition of length at most m (see Appendix A).

Then we have the following fact.

Proposition 2. The polynomial det ( ∑_{a=1}^{k} u_{i}^{a}v_{j}^{a} P (z⃗_{\i}, y_{j}))_{1≤i,j≤n} is divisible by the

polynomial (P (z⃗))^{n−k}.

Proof. Apply the formula for the minor expansion of a sum of matrices given in Proposition 1, to get

det ( ∑_{a=1}^{k} u_{i}^{a}v_{j}^{a}P (z⃗_{\i}, y_{j}))_{1≤i,j≤n}

= ∑_{\begin{matrix}I,J \\ I∼J \end{matrix}} ϵ(I, J) ∏_{\begin{matrix}1≤a≤k \\ i∈I_{a} \end{matrix}}u_{i}^{a} ∏_{\begin{matrix}1≤a≤k \\ j∈J_{a} \end{matrix}}v_{j}^{a} ∏_{a=1}^{k} det (P (z⃗_{\i}, y_{j}))_{i∈I_{a}, j∈J_{a}} .

To each determinant of the form det (P (z⃗_{\i}, y_{j}))_{i∈I, j∈J} apply the Bazin–Reiss–Picquet Theorem with (m, n, p) → (m, |I|, 1), and get

det (P (z⃗_{\i}, y_{j}))_{i∈I, j∈J} = P (z⃗)^{|I|−1}P (w⃗(I, J)) ,

where

w_{k}(I, J) = \begin{cases}z_{k} & if k ∉ I; \\ y_{j_{ℓ}} & if k = i_{ℓ}. \end{cases}

Thus we have

det ( ∑_{a=1}^{k} u_{i}^{a}v_{j}^{a}P (z⃗_{\i}, y_{j}))_{1≤i,j≤n} = P (z⃗)^{n−k} ∑_{\begin{matrix}I,J \\ I∼J \end{matrix}} ϵ(I, J) ∏_{\begin{matrix}1≤a≤k \\ i∈I_{a} \end{matrix}}u_{i}^{a} ∏_{\begin{matrix}1≤a≤k \\ j∈J_{a} \end{matrix}}v_{j}^{a} ∏_{a=1}^{k} P (w⃗(I_{a}, J_{a})) ,

and the quantity in the sum on the right-hand side is a polynomial. □

3.4. Vanishing and recursion properties of 2-staircase Schur functions. Here we gather some relevant facts about the family of 2-staircase Schur functions s_{λ_{n,ℓ,ℓ′}}(z⃗) introduced in (1.3). In this section we use q as synonym for exp(\frac{2πi}{ℓ+2}).

Proposition 3 (Wheel condition). For distinct g, h and k in {0, …, ℓ + 1}, and distinct i, j and m in {1, …, 2n}, we have

(3.4)  s_{λ_{n,ℓ,ℓ′}}(z⃗_{\ijm}, q^{g}w, q^{h}w, q^{k}w) = 0 .

Proposition 4 (Recursion relation). For k in {1, …, ℓ + 1}, and distinct i, j in {1, …, 2n}, we have

(3.5)  s_{λ_{n,ℓ,ℓ′}}(z⃗_{\ij}, w, q^{k}w) = w^{ℓ′}U_{ℓ′}(1, q^{k}) ∏_{\begin{matrix}1≤m≤2n \\ m≠i,j \end{matrix}} \frac{U_{ℓ+1}(z_{m}, w)}{z_{m} − q^{k}w} s_{λ_{n−1,ℓ,ℓ′}}(z⃗_{\ij}) .

Propositions 3 and 4 are occurrences, already known in the literature (cf., e.g., [18, Thm. 4]), of vanishing conditions (and related recursion properties) within a broad family, for which the name “wheel condition” is often used. There has been a recent interest in the investigation of the structure of the corresponding ideals, in the ring of symmetric polynomials (see, e.g., [3, 4]).

We prove the propositions above in Appendix B. More precisely, in the appendix we generalize 2-staircase Schur functions to the m-staircase case, and prove the appropriate generalizations of the propositions above, together with some further properties of potential future interest.

Notice that, if gcd(ℓ′ +1, ℓ+2) = g > 1, then there exists some k with 1 ≤ k ≤ ℓ+1 such that q^{k} is a root of U_{ℓ′}(1, x) (e.g., k = (ℓ+2)/g). Then it follows from equation (3.5) that s_{λ_{n,ℓ,ℓ′}} vanishes if z_{i} = q^{k}z_{j}, i.e., it is divisible by z_{i} − q^{k}z_{j}. On the other hand, if gcd(ℓ′ + 1, ℓ + 2) = 1, one has the following proposition.

Proposition 5. Suppose gcd(ℓ′ + 1, ℓ + 2) = 1 and n ≥ 2, then s_{λ_{n,ℓ,ℓ′}} has no factors of the form (z_{i} − ηz_{j}), for any ij with 1 ≤ i, j ≤ 2n and η ∈ ℂ.

Proof. We prove the statement by induction on n. The case n = 2 is done by direct inspection of s_{λ_{n,ℓ,ℓ′}}.^{1} Now suppose the statement is true up to n − 1 and assume that there exists i, j ∈ {1, …, 2n} and η ∈ ℂ such that (z_{i} − ηz_{j}) divides s_{λ_{n,ℓ,ℓ′}}. Then take k and h distinct indices in {1, …, 2n}\{i, j} (note that we need n ≥ 2 at this point), and specialize s_{λ_{n,ℓ,ℓ′}}|_{z_{k}=qz_{h}}. The linear term z_{i} − ηz_{j} must divide also the specialized polynomial, and, using the recursion relation of Proposition 4, it must divide the corresponding right-hand side expression for (3.5). However, this expression is nonzero for the other variables z_{m} being generic (because the only potentially dangerous factor, U_{ℓ′}(1, q^{k}), may vanish only if gcd(ℓ′ + 1, ℓ + 2) > 1), and the factors of the form z_{k}^{ℓ′}, and U_{ℓ+1}(z_{m}, z_{k}), for m ≠ k, h, do not contain z_{i} − ηz_{j} as a factor. Thus z_{i} − ηz_{j} must divide s_{λ_{n−1,ℓ,ℓ′}}, in contradiction with the inductive hypothesis. □

1E.g., observe that, for z_{1} − ηz_{2} to divide the Schur function, it should divide the shifted Vandermonde determinant in the numerator, with a higher power with respect to the ordinary Vandermonde determinant in the denominator. The case η = 1 is easily ruled out (even if we further specialize z_{3} = z, z_{4} = 0, we obtain s_{λ_{2,ℓ,ℓ′}} (z, z, z, 0) = z^{2(ℓ+ℓ′)}(ℓ + 2)(ℓ′ + 1)(ℓ − ℓ′ + 1)/2, which is not identically zero as we have ℓ ≥ 0 and 0 ≤ ℓ′ ≤ ℓ). For η ≠ 1 we cannot have simplifications with the Vandermonde determinant in the denominator, and it suffices to analyse the shifted Vandermonde determinant, which gives

Δ_{λ_{2,ℓ,ℓ′}} (z, ηz, 0, 1) = z^{ℓ+ℓ′+3}(((ηz)^{ℓ+2} − 1)(η^{ℓ′+1} − 1) − ((ηz)^{ℓ′+1} − 1)(η^{ℓ+2} − 1)) .

Again, this is not identically zero, as, because of the gcd hypothesis, η^{ℓ′+1} − 1 and η^{ℓ+2} − 1 cannot vanish simultaneously.

4. Proof of Theorem 2

As outlined in the introduction, our strategy for proving Theorem 2 will be as follows: let us call ψ_{n,ℓ,ℓ′}(z, x, y) the left-hand side of (1.7); first we identify several polynomial factors of ψ_{n,ℓ,ℓ′}(z, x, y); then we show that these factors are relatively prime and that their product exhausts the degree of ψ_{n,ℓ,ℓ′}(z, x, y); finally, we determine the overall constant factor. As in the previous subsection, also in this section we set q = e^{\frac{2πi}{ℓ+2}} .

4.1. Polynomial factors of ψ_{n,ℓ,ℓ′}(z⃗, x⃗, y⃗). We start by identifying a polynomial factor of ψ_{n,ℓ,ℓ′}(z⃗, x⃗, y⃗) whose factorization involves only monomials and binomials. By Lemma 1, the polynomial ψ_{n,ℓ,ℓ′}(z⃗, x⃗, y⃗) is divisible by Δ(x⃗) and Δ(y⃗). Since the degree of ψ_{n,ℓ,ℓ′} in each variable x_{i} or y_{i} separately is (n − 1)ℓ + ℓ′, which is the same as the degree of Δ(x⃗)Δ(y⃗), the quotient is a polynomial of degree zero in x_{i} and y_{j} (namely, it is the determinant of the matrix of coefficients in x and y of s_{λ_{n,ℓ,ℓ′}}(z⃗, x, y)). Call Q_{n,ℓ,ℓ′}(z⃗) the resulting quotient

(4.1)  Q_{n,ℓ,ℓ′}(z⃗) = \frac{ψ_{n,ℓ,ℓ′}(z⃗, x⃗, y⃗)}{Δ(x⃗)Δ(y⃗)}.

We work out immediately the case of Theorem 2 corresponding to the second case of equation (1.8)

Proposition 6. If gcd(ℓ′ + 1, ℓ + 2) > 1 and n ≥ 2, then Q_{n,ℓ,ℓ′}(z⃗) = 0.

Proof. Say gcd(ℓ′ + 1, ℓ + 2) = g > 1. It follows that the polynomials U_{ℓ′}(1, x) and U_{ℓ+1}(1, x) have a common root q^{k}, for k = (ℓ+2)/g. We can exploit the fact that Q, defined in equation (4.1) as a rational function of the z, x and y’s, is actually independent of the x and y’s. In particular, we can choose x_{1} = q^{k}z_{1} (and leave x_{2}, …, x_{n}, y_{1}, …, y_{n} generic). Consider the matrix (M_{ij})_{1≤i,j≤N} with M_{ij} = s_{λ_{n,ℓ,ℓ′}}(z⃗, x_{i}, y_{j}), whose determinant is ψ_{n,ℓ,ℓ′}. By applying the recursion relation of Proposition 4, we see that the row corresponding to x_{1} vanishes identically. On the other hand, as the remaining x and y variables are generic, the Vandermonde factors are non-zero. As a consequence, Q_{n,ℓ,ℓ′}(z⃗) = 0. □

We proceed to find other factors of Q_{n,ℓ,ℓ′}, for the relevant case of equation (1.8).

Proposition 7. For n ≥ 2, s_{μ_{2n−2,ℓ+1}}^{ℓ}(z⃗)( ∏_{i=1}^{2n−2} z_{i}^{ℓ′(ℓ+1)}) divides Q_{n,ℓ,ℓ′}(z⃗).

Proof. Note that Q_{n,ℓ,ℓ′}(z⃗) is symmetric in the z_{i}’s (as they enter only as simultaneous arguments of Schur functions). So, given the factorized form of s_{μ}, equation (1.10), it suffices to prove that Q is divisible by z_{1}^{ℓ′(ℓ+1)} ∏_{m=2}^{2n−2} U_{ℓ+1}^{ℓ}(z_{1}, z_{m}). Using the independence from x⃗ and y⃗ of equation (4.1), we can choose to substitute x_{i} = q^{i}z_{1} for 1 ≤ i ≤ ℓ + 1 and leave generic the other x_{j}’s and all the y_{j}’s (we have a sufficient number of x’s since (n − 1)ℓ + ℓ′ + 1 ≥ ℓ + 1 for n ≥ 2).

By applying the recursion relation of Proposition 4 to the matrix entries M_{ij}, the first ℓ + 1 rows of M are simplified. Consider the matrix M̃ that coincides with M in rows i > ℓ + 1, and otherwise is given by

(4.2)  M̃_{ij} = (z_{1}^{ℓ′}U_{ℓ′}(1, q^{i}) ∏_{m=2}^{2n−2} \frac{U_{ℓ+1}(z_{m}, z_{1})}{z_{m} − q^{i}z_{1}})(\frac{U_{ℓ+1}(y_{j}, z_{1})}{y_{j} − x_{i}} s_{λ_{n−1,ℓ,ℓ′}}(z⃗_{\1}, y_{j})).

This matrix is a version of M in which we do not replace x_{i} → q^{i}z_{1} for all the occurrences of x_{i} in M_{ij}, but only for a subset. That is, we just have the property,

M_{ij} = M̃_{ij}|_{x_{i}=q^{i}z_{1}} , for 1 ≤ i ≤ ℓ + 1,

and thus det M = (det M̃)|_{x_{i}=q^{i}z_{1}}. We constructed M̃ instead of M with specific intentions: the two factors in parenthesis in (4.2) are separately polynomials after replacing x_{i} = q^{i}z_{1} (and, before the replacement, they are at most divisible by y_{j} − x_{i}); the factor on the left does not depend on index j (so it can be extracted from the i-th row of M̃ when evaluating the determinant); finally, the dependence from i in the second factor is all due to x_{i} (so that the i-th and i′-th row of M are the same vector of functions, with different x argument, i.e., det M̃ is visibly divisible by Δ(x_{1}, …, x_{ℓ+1})).

The factors extracted from the rows give

∏_{i=1}^{ℓ+1} (z_{1}^{ℓ′}U_{ℓ′}(1, q^{i}) ∏_{2≤m≤2n−2} \frac{U_{ℓ+1}(z_{m}, z_{1})}{z_{m} − q^{i}z_{1}}) ,

that is, with some simplifications (including ∏_{i=1}^{ℓ+1} U_{ℓ′}(1, q^{i}) = 1 if gcd(ℓ + 2, ℓ′ + 1) = 1 and 0 otherwise),

(4.3)  z_{1}^{ℓ′(ℓ+1)} ∏_{m=2}^{2n−2} U_{ℓ+1}^{ℓ}(z_{1}, z_{m}) .

The divisibility of det M̃ by Δ(x_{1}, …, x_{ℓ+1}) implies that det M̃/Δ(x_{1}, …, x_{N}) has no factors x_{i} − x_{i′} in the denominator with 1 ≤ i < i′ ≤ ℓ + 1, and thus no pure powers of z_{1} at the denominator from the Vandermonde determinant, after the replacement x_{i} = q^{i}z_{1} (indeed, all the potential factors in the denominator have the form q^{i}z_{1} − x_{j}, with j > ℓ + 1, and y_{j} − q^{i}z_{1}, with j ≤ ℓ + 1), thus they do not affect the claimed factor in (4.3). This completes the proof. □

Now we complete the exhaustion of factors, by proving the following weaker form of Theorem 2.

Proposition 8. For n ≥ 2 and ℓ ≥ 1 we have

(4.4)  Q_{n,ℓ,ℓ′}(z⃗) = c(n, ℓ, ℓ′)( ∏_{i=1}^{2n−2} z_{i}^{ℓ′(ℓ+1)}) s_{μ_{2n−2,ℓ+1}}^{ℓ}(z⃗) s_{λ_{n−1,ℓ,ℓ′}}^{ℓ(n−2)+ℓ′−1}(z⃗) ,

for some numerical constant c(n, ℓ, ℓ′).

Proof. As a consequence of Proposition 6, our claim is trivially true if gcd(ℓ′ + 1, ℓ + 2) > 1, as the constant in such a case is 0. Therefore it remains to analyse the case gcd(ℓ′ + 1, ℓ + 2) = 1.

We can again exploit the invariance in x and y of Q_{n,ℓ,ℓ′}(z⃗) from equation (4.1), in order to evaluate ψ_{n,ℓ,ℓ′}(z⃗, x⃗, y⃗) at a special set of values x and y. Our choice is to leave the y_{j}’s generic, and specialize x_{i} = q^{k_{i}}z_{m_{i}}, for all the indices i = 1, …, N (recall that we defined N = ℓ(n + 1) + ℓ′ + 1), and {(k_{i}, m_{i})} being a whatever ordered subset of distinct pairs, of cardinality N, in the set of all valid pairs {1, …, ℓ+1}×{1, …, 2n−2}. The difference of cardinality, (ℓ + 2)(n − 1) − ℓ′ − 1, is always positive in our range of interest ℓ ≥ 1, 0 ≤ ℓ′ ≤ ℓ, n ≥ 2. Using Proposition 4, we have

M_{ij} = s_{λ_{n,ℓ,ℓ′}}(z⃗, x_{i} = q^{k_{i}}z_{m_{i}}, y_{j})

= z_{m_{i}}^{ℓ′}U_{ℓ′}(1, q^{k_{i}})\frac{U_{ℓ+1}(y_{j}, z_{m_{i}})}{y_{j} − q^{k_{i}}z_{m_{i}}} ∏_{\begin{matrix}1≤r≤2n−2 \\ r≠m_{i} \end{matrix}} \frac{U_{ℓ+1}(z_{r}, z_{m_{i}})}{z_{r} − q^{k_{i}}z_{m_{i}}}s_{λ_{n−1,ℓ,ℓ′}}(z⃗_{\m_{i}}, y_{j}) .

Let us adopt the representation (A.1) for the Schur polynomial (as the ratio of shifted Vandermonde determinant by a Vandermonde determinant), to get

(4.5) M_{ij} = \frac{z_{m_{i}}^{ℓ′}U_{ℓ′}(1, q^{k_{i}})}{Δ(z⃗_{\m_{i}}, y_{j})} \frac{U_{ℓ+1}(y_{j}, z_{m_{i}})}{y_{j} − q^{k_{i}}z_{m_{i}}} ∏_{\begin{matrix}1≤r≤2n−2 \\ r≠m_{i} \end{matrix}} \frac{U_{ℓ+1}(z_{r}, z_{m_{i}})}{z_{r} − q^{k_{i}}z_{m_{i}}} Δ_{λ_{n−1,ℓ,ℓ′}}(z⃗_{\m_{i}}, y_{j})

= \frac{z_{m_{i}}^{ℓ′}U_{ℓ′}(1, q^{k_{i}})}{Δ(z⃗)}(−1)^{m_{i}−1}( ∏_{r≠m_{i}} \frac{(z_{r} − z_{m_{i}})U_{ℓ+1}(z_{r}, z_{m_{i}})}{z_{r} − q^{k_{i}}z_{m_{i}}})( ∏_{r} \frac{1}{y_{j} − z_{r}} )

× \frac{(y_{j} − z_{m_{i}})U_{ℓ+1}(y_{j}, z_{m_{i}})}{y_{j} − q^{k_{i}}z_{m_{i}}} Δ_{λ_{n−1,ℓ,ℓ′}}(z⃗_{\m_{i}}, y_{j})

= \frac{z_{m_{i}}^{ℓ′}U_{ℓ′}(1, q^{k_{i}})}{Δ(z⃗)}((−1)^{m_{i}−1} ∏_{r≠m_{i}} U_{ℓ+1}(z_{r}, q^{k_{i}}z_{m_{i}}))( ∏_{r} \frac{1}{y_{j} − z_{r}} )

× U_{ℓ+1}(y_{j}, q^{k_{i}}z_{m_{i}}) Δ_{λ_{n−1,ℓ,ℓ′}}(z⃗_{\m_{i}}, y_{j}) ,

where in the last equality we made use of the relation

(4.6)  \frac{U_{ℓ+1}(x, q^{h}y)}{x − q^{k}y} = ∏_{\begin{matrix}0≤i≤ℓ+1 \\ i≠h,k \end{matrix}}(x − q^{i}y) = \frac{U_{ℓ+1}(x, q^{k}y)}{x − q^{h}y} .

In the last expression of equation (4.5), we recognize five factors: a factor independent of i and j, one depending on i alone, one depending on j alone, and one depending on both i and j, which is composed of U_{ℓ+1}(y_{j}, q^{k_{i}}z_{m_{i}}), which is a homogeneous polynomial in y_{j} and z_{m_{i}} of degree ℓ + 1, and a shifted Vandermonde determinant. The first three factors are easily extracted when evaluating det M, so we can write

(4.7)  det M = \frac{A(z⃗, y⃗)}{B(z⃗, y⃗)Δ(z⃗)^{N}} det M̂

with

(4.8)  M̂_{ij} = U_{ℓ+1}(y_{j}, q^{k_{i}}z_{m_{i}})Δ_{λ_{n−1,ℓ,ℓ′}}(z⃗_{\m_{i}}, y_{j}) ;
(4.9)  A(z⃗, y⃗) = ∏_{i}(−1)^{m_{i}−1}z_{m_{i}}^{ℓ′}U_{ℓ′}(1, q^{k_{i}}) ∏_{\begin{matrix}1≤i≤N \\ 1≤r≤2n−2 \\ r≠m_{i} \end{matrix}}U_{ℓ+1}(z_{r}, q^{k_{i}}z_{m_{i}}) ;

(4.10)  B(z⃗, y⃗) = ∏_{\begin{matrix}1≤j≤N \\ 1≤r≤2n−2 \end{matrix}}(y_{j} − z_{r}) .

We now substitute the expression of equation (4.7) in (4.1), where we also do the replacement

Δ(x⃗) ⟶ Δ(q^{k_{1}}z_{m_{1}}, q^{k_{2}}z_{m_{2}}, …) ,

which leads to

(4.11)  Q_{n,ℓ,ℓ′}(z⃗) = \frac{A(z⃗, y⃗)}{B(z⃗, y⃗)} \frac{1}{Δ(y⃗)Δ(q^{k_{1}}z_{m_{1}}, q^{k_{2}}z_{m_{2}}, …)Δ(z⃗)^{N}} det M̂.

Now, the matrix M̂ is in a form suitable for application of Proposition 2, the divisibility result discussed in Section 3.3, with k = ℓ + 2 and, for 0 ≤ a ≤ ℓ + 1, u_{i}^{a}v_{j}^{a} being the coefficient of the monomial y_{j}^{a}z_{m_{i}}^{ℓ+1−a} in the expansion of U_{ℓ+1}(y_{j}, q^{k_{i}}z_{m_{i}}).

As a consequence, the polynomial Δ_{λ_{n−1,ℓ,ℓ′}}^{N−(ℓ+2)}(z⃗) divides det M̂, and the exponent N − (ℓ + 2) = ℓ(n − 1) + ℓ′ + 1 − (ℓ + 2) = ℓ(n − 2) + ℓ′ − 1 is exactly the desired one from the statement of Proposition 8 (and Theorem 2). So we can write

det M̂ = Δ_{λ_{n−1,ℓ,ℓ′}}^{ℓ(n−2)+ℓ′−1}(z⃗) R(z⃗, y⃗)

for R a polynomial. If we substitute this in (4.11), then we obtain

(4.12)  Q_{n,ℓ,ℓ′}(z⃗) = s_{λ_{n−1,ℓ,ℓ′}}^{ℓ(n−2)+ℓ′−1}(z⃗) \frac{A(z⃗, y⃗)R(z⃗, y⃗)}{B(z⃗, y⃗)Δ(y⃗)Δ(q^{k_{1}}z_{m_{1}}, q^{k_{2}}z_{m_{2}}, …)Δ(z⃗)^{ℓ+2}} .

Now, as gcd(ℓ′ + 1, ℓ + 2) = 1, we obtain two consequences from Proposition 5. First, observing that the denominator in (4.12) is completely factorized into linear terms (of the form y_{i} − z_{j}, or z_{i} − q^{k}z_{j}), s_{λ_{n−1,ℓ,ℓ′}}(z⃗) cannot be divisible by any of these factors, therefore it follows from equation (4.12) that s_{λ_{n−1,ℓ,ℓ′}}^{ℓ(n−2)+ℓ′−1}(z⃗) must divide Q_{n,ℓ,ℓ′}(z⃗).

Furthermore, we know from Proposition 7 that s_{μ_{2n−2,ℓ+1}}^{ℓ}(z⃗) ∏_{i=1}^{2n−2} z_{i}^{ℓ′(ℓ+1)} divides Q_{n,ℓ,ℓ′}(z⃗). Also this polynomial is factorized into linear terms, of the form z_{i} or z_{i} −q^{k}z_{j}, thus it is relatively prime with s_{λ_{n−1,ℓ,ℓ′}}^{ℓ(n−2)+ℓ′−1}. This shows that Proposition 8 holds, for c(n, ℓ, ℓ′) a polynomial. However, all the involved functions are homogeneous polynomials, and it is easily determined that c(n, ℓ, ℓ′) has degree 0, thus it is a constant. □

4.2. Determination of the constant c(n, ℓ, ℓ′). We can evaluate directly the

constant for n = 1, which is c(1, ℓ, ℓ′) = (−1)^{\binom{ℓ′+1}{2}}, and we know that, for n ≥ 2 and gcd(ℓ + 2, ℓ′ + 1) > 1, c(n, ℓ, ℓ′) = 0. In the rest of this section we will complete the proof of expression (1.8), for the remaining case n ≥ 2 and gcd(ℓ + 2, ℓ′ + 1) = 1. This is done by induction in n, i.e., we will prove that, for (n, ℓ, ℓ′) as above,

\frac{c(n, ℓ, ℓ′)}{c(n − 1, ℓ, ℓ′)} = (−1)^{\binom{ℓ+1}{2}} .

Now that we only have to determine the constant, we have the freedom of choosing simpler values also for the z_{k}’s, in addition to the ones for the x_{i}’s and the y_{j}’s.

First of all, in equation (4.1) let us specialize x_{i} = q^{i}z_{1} for 1 ≤ i ≤ ℓ. In this way we find that the matrix entries M_{ij} for 1 ≤ i ≤ ℓ take the form^{2}

(4.13)  M_{ij} = z_{1}^{ℓ′}U_{ℓ′}(1, q^{i})\frac{U_{ℓ+1}(y_{j}, z_{1})}{y_{j} − q^{i}z_{1}} ∏_{r=2}^{2n−2} \frac{U_{ℓ+1}(z_{r}, z_{1})}{z_{r} − q^{i}z_{1}}s_{λ_{n−1,ℓ,ℓ′}}(z⃗_{\1}, y_{j}).

As we have done in the proof of Proposition 7, when we compute the determinant of the matrix M, for 1 ≤ i ≤ ℓ we extract the factor

(4.14)  z_{1}^{ℓ′}U_{ℓ′}(1, q^{i}) ∏_{r=2}^{2n−2} \frac{U_{ℓ+1}(z_{r}, z_{1})}{z_{r} − q^{i}z_{1}}

from the i-th row, and find that

(4.15)  det M = F (z_{1}; z⃗_{\1}) det M′,

where

F (z_{1}; z⃗_{\1}) = \frac{z_{1}^{ℓ′ℓ}}{U_{ℓ′}(1, q^{ℓ+1})} ∏_{r=2}^{2n−2} U_{ℓ+1}^{ℓ−1}(z_{r}, z_{1}) (z_{r} − q^{ℓ+1}z_{1}),

and the matrix M′ coincides with M in the last N − ℓ rows, while each of the first ℓ rows is divisible by the factor in equation (4.14).

We now substitute the expression (4.15) for det M into the definition of Q_{n,ℓ,ℓ′}(z⃗) and then into equation (4.4), taking into account also the substitutions of the variables in the Vandermonde determinant in the denominator

(4.16)  Δ(x⃗) ⟶ z\binom{ℓ}{2}_{1}Δ′(z_{1}, x⃗_{\1,...ℓ}),
(4.17)  Δ′(z_{1}, x⃗_{\1,...ℓ}) := Δ(q, q^{2}, …, q^{ℓ}) Δ(x⃗_{\1,...ℓ}) ∏_{\begin{matrix}1≤i≤ℓ \\ ℓ+1≤k≤N \end{matrix}}(q^{i}z_{1} − x_{k}) ,

(4.18)  Δ(q, q^{2}, …, q^{ℓ})^{2} = (−1)^{\binom{ℓ+1}{2}}(q^{−1} − q^{−2})^{2}(ℓ + 2)^{ℓ−2} .

We obtain

(4.19)  Q_{n,ℓ,ℓ′}(z⃗) = \frac{F (z_{1}; z⃗_{\1}) det M′}{z\binom{ℓ}{2}_{1}Δ′(z_{1}, x⃗_{\1,...ℓ})Δ(y⃗)}

= c(n, ℓ, ℓ′) ∏_{i=1}^{2n−2} z_{i}^{ℓ′(ℓ+1)} s_{μ_{2n−2,ℓ+1}}^{ℓ}(z⃗) s_{λ_{n−1,ℓ,ℓ′}}^{ℓ(n−2)+ℓ′−1}(z⃗) .

2That is, nothing else but M̃_{ij} in (4.2), under the full replacement x_{i} → q^{i}z_{1}.

We eliminate the factors appearing on both sides of the previous equation, and we obtain

(4.20) \frac{det M′}{U_{ℓ′}(1, q^{ℓ+1})Δ′(z_{1}, x⃗_{\1,...ℓ})Δ(y⃗)} = c(n, ℓ, ℓ′)z_{1}^{\binom{ℓ}{2}+ℓ′}

× ∏_{i=2}^{2n−2} z_{i}^{ℓ′(ℓ+1)} ∏_{r=2}^{2n−2} \frac{U_{ℓ+1}(z_{r}, q^{ℓ+1}z_{1})}{z_{r} − z_{1}}s_{μ_{2n−3,ℓ+1}}^{ℓ}(z⃗_{\1}) s_{λ_{n−1,ℓ,ℓ′}}^{ℓ(n−2)+ℓ′−1}(z⃗) .

Note that, among other things, we have eliminated all the factors z_{r} − q^{ℓ+1}z_{1} on both sides. This allows us to set z_{2} = q^{ℓ+1}z_{1}. Furthermore, we choose to specialize y_{i} = q^{i}z_{1}, for 1 ≤ i ≤ ℓ (the Vandermonde factor Δ(y⃗) in equation (4.20) is then to be treated similarly to what is done in (4.16) for Δ(x⃗)).

It is easy to see which simplifications occur on the factorized right-hand side of equation (4.20)

(4.21)  ∏_{i=2}^{2n−2} z_{i}^{ℓ′(ℓ+1)} → q^{ℓ′}z_{1}^{ℓ′(ℓ+1)} ∏_{i=3}^{2n−2} z_{i}^{ℓ′(ℓ+1)}

(4.22)  ∏_{r=2}^{2n−2} \frac{U_{ℓ+1}(z_{r}, q^{ℓ+1}z_{1})}{z_{r} − z_{1}} → \frac{z_{1}^{ℓ}(ℓ + 2)}{q^{−2} − q^{−1}} ∏_{r=3}^{2n−2} \frac{U_{ℓ+1}(z_{r}, q^{ℓ+1}z_{1})}{z_{r} − z_{1}}

(4.23)  s_{μ_{2n−3,ℓ+1}}^{ℓ}(z⃗_{\1}) → ∏_{r=3}^{2n−2} U_{ℓ+1}^{ℓ}(z_{r}, q^{ℓ+1}z_{1}) s_{μ_{2n−4,ℓ+1}}^{ℓ}(z⃗_{\1,2})

(4.24)  s_{λ_{n−1,ℓ,ℓ′}}(z⃗) → z_{1}^{ℓ′}U_{ℓ′}(1, q^{ℓ+1}) ∏_{r=3}^{2n−2} \frac{U_{ℓ+1}(z_{r}, z_{1})}{z_{r} − q^{ℓ+1}z_{1}}s_{λ_{n−2,ℓ,ℓ′}}(z⃗_{\1,2}) .

Even more drastic simplifications arise on the left-hand side of equation (4.20). For i > ℓ and j ≤ ℓ, the entries M′_{ij} consist of the Schur polynomials s_{λ_{n,ℓ,ℓ′}} evaluated at a set of variables including a triple satisfying the wheel condition (namely, z_{1}, y_{j} = q^{j}z_{1} and z_{2} = q^{ℓ+1}z_{1}), therefore they vanish because of Proposition 3. Similarly, for i ≤ ℓ and j ≤ ℓ, with the only exception of i = j, M′_{ij} vanishes because of the factor \frac{U_{ℓ+1}(y_{j},z_{1})}{y_{j}−q^{i}z_{1}} = ∏_{1≤k≤ℓ+1; k≠i}(y_{j} − q^{k}z_{1}) (cf. equation (4.13)). As a result,

(4.25)  det M′ = ( ∏_{i=1}^{ℓ} M′_{ii}) det M′_{{ℓ+1,...,N},{ℓ+1,...,N}}.

The diagonal factors M′_{ii} read

M′_{ii} = \frac{z_{1}^{ℓ+ℓ′}(ℓ + 2)q^{ℓi}U_{ℓ′}(q^{ℓ+1}, q^{i})}{1 − q^{−i}} ∏_{r=3}^{2n−2} \frac{U_{ℓ+1}(z_{r}, q^{ℓ+1}z_{1})}{z_{r} − q^{i}z_{1}}s_{λ_{n−2,ℓ,ℓ′}}(z⃗_{\1,2}).

Most importantly, the minor of the matrix M′ restricted to the last N − ℓ rows and columns is easily related to the matrix M for the system of size n−1, where the indices of the variables z_{k} run from 3 to 2n − 2, while the indices of the x_{i}’s and y_{j}’s run from ℓ + 1 to N = (n − 1)ℓ + ℓ′ + 1. More precisely, M′_{ℓ+i,ℓ+j}, at size n and under the specializations above, is proportional to M_{ij} at size n − 1, the proportionality factor for the pair (i, j) being

(4.26)  z_{1}^{ℓ′}U_{ℓ′}(1, q^{ℓ+1})( ∏_{r=3}^{2n−2} \frac{U_{ℓ+1}(z_{r}, z_{1})}{z_{r} − q^{ℓ+1}z_{1}})\frac{U_{ℓ+1}(x_{ℓ+i}, z_{1})}{x_{ℓ+i} − q^{ℓ+1}z_{1}} \frac{U_{ℓ+1}(y_{ℓ+j}, z_{1})}{y_{ℓ+j} − q^{ℓ+1}z_{1}}.

(The relevant fact is that this quantity factorizes into a term depending on x_{i} only, and a term depending on y_{j} only, these terms thus factorize in the evaluation of the determinant.) Thus we get

det M′_{\{ℓ+1,...,N\},\{ℓ+1,...,N\}} = [z_{1}^{ℓ′}U_{ℓ′}(1, q^{ℓ+1})( ∏_{r=3}^{2n−2} \frac{U_{ℓ+1}(z_{r}, z_{1})}{z_{r} − q^{ℓ+1}z_{1}})]^{N−ℓ} (4.27)

× ∏_{i=1}^{N−ℓ} \frac{U_{ℓ+1}(x_{ℓ+i}, z_{1})}{x_{ℓ+i} − q^{ℓ+1}z_{1}} ∏_{j=1}^{N−ℓ} \frac{U_{ℓ+1}(y_{ℓ+j}, z_{1})}{y_{ℓ+j} − q^{ℓ+1}z_{1}}

× Δ(x_{ℓ+1}, …, x_{N}) Δ(y_{ℓ+1}, …, y_{N}) Q_{n−1,ℓ,ℓ′}(z_{3}, …, z_{2n}) .

In this equation we can replace Q_{n−1,ℓ,ℓ′}(z⃗_{\1,2}) by its expression given by equation (4.4)

— the factor c(n − 1, ℓ, ℓ′) emerges at this point — then, we can substitute (4.27) in (4.20), using (4.25). In this way we arrive at a fully factorized form on both sides of equation (4.20). Cancellation of common factors is a matter of simple algebra,^{3}which, in the end, leads to the recurrence relation

c(n, ℓ, ℓ′) = (−1)^{\binom{ℓ+1}{2}}c(n − 1, ℓ, ℓ′) ,

as was to be proven. □

Appendix A. Basic facts on symmetric polynomials

A partition λ is a (finite or infinite) non-increasing sequence of non-negative integers, λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k} ⋯) containing only finitely many non-zero terms. The number of non-zero terms (or parts) ℓ(λ), and the value of the sum |λ| = ∑_{i=1}^{k} λ_{i}, are called the length and the weight of the partition, respectively. When there is no confusion, we use the same name λ for the finite partition (λ_{1}, λ_{2}, …, λ_{k}) and for the infinite one (λ_{1}, λ_{2}, …, λ_{k}, 0, 0, …). Seen as an arrangement of cells (as, e.g., in Figure 2), λ is often called a Young diagram.

Given an ordered ℓ-tuple of indeterminates z⃗ = {z_{i}}_{1≤i≤ℓ}, the Vandermonde determinant Δ(z⃗) is defined as the determinant of the ℓ × ℓ matrix V = (V_{ij})_{1≤i,j≤ℓ} with V_{ij} = z_{i}^{ℓ−j}. It is well known that Δ(z⃗) = ∏_{1≤i<j≤ℓ}(z_{i}−z_{j}). For a partition λ of length at most ℓ, one similarly defines the shifted Vandermonde determinant Δ_{λ}(z⃗) as the determinant of the ℓ×ℓ matrix V = (V_{ij})_{1≤i,j≤ℓ} with V_{ij} = z_{i}^{λ_{j}+ℓ−j}. Thus Δ(z⃗) ≡ Δ_{(0,0,...,0)}(z⃗).

∏_{i=1}^{ℓ} \frac{q^{ℓi}}{1 − q^{−i}} = \frac{q^{−2} − q^{−1}}{ℓ + 2} ;  ∏_{i=1}^{ℓ} U_{ℓ′}(q^{ℓ+1}, q^{i}) = \frac{q^{ℓ′}}{U_{ℓ′}(1, q^{ℓ+1})} .

3Useful relations at this point are (4.18) and

Then the Schur polynomial associated to λ is the function in ℓ indeterminates

(A.1)  s_{λ}(z⃗) = \frac{Δ_{λ}(z⃗)}{Δ(z⃗)} .

It is indeed a polynomial, it is symmetric in all its variables, and homogeneous of degree |λ|. The Schur functions are at the heart of algebraic combinatorics [15] and enjoy several remarkable properties (see [10]). Here we limit ourselves to presenting the few simple results we need in the paper, among which the following “splitting formula”.

Proposition 9. Let (λ_{1}, …, λ_{k}) and (μ_{1}, …, μ_{h}) be two partitions such that λ_{k} ≥ μ_{1}. Let ν denote the partition ν = (λ_{1}, …, λ_{k}, μ_{1}, …, μ_{h}). Then we have

(A.2)  lim_{ϵ→0} \frac{s_{ν}(z_{1}, …, z_{k}, ϵy_{1}, …, ϵy_{h})}{ϵ^{|μ|}} = s_{λ}(z_{1}, …, z_{k}) s_{μ}(y_{1}, …, y_{h}) .

This generalizes the simple property that a Schur polynomial s_{λ}(z⃗) in ℓ variables has maximum degree λ_{1}, and mimimum degree λ_{ℓ}, in any of its variables. For the connoisseurs, the proposition can easily be proven in several ways, for example by using the decomposition formula for Schur function s_{α}(x⃗, y⃗) = ∑_{β⊆α} s_{β}(x⃗)s_{α/β}(y⃗) (see, e.g., [10, eq. (5.9)]), and simple properties of skew Schur functions (which we do not introduce). Here we provide a more verbose but completely self-contained proof.

Proof. Using the defining equation (A.1), we are led to study the behaviour of Δ_{γ}(z⃗, ϵy⃗) as ϵ → 0, for the cases γ = ν (in the numerator) and γ = 0 (in the denominator). More generally, consider γ = (γ_{1}, …, γ_{k+h}) ≡ (α_{1}, …, α_{k}, β_{1}, …, β_{h}). Recall that Δ_{γ}(z⃗, ϵy⃗) is defined as the determinant of the matrix V = (V_{ij})_{1≤i,j≤k+h} with V_{ij} = z_{i}^{γ_{j}+k+h−j} for i ≤ k and V_{ij} = (ϵy_{i−k})^{γ_{j}+k+h−j} for i > k. Consider the Laplace expansion of V along the first k rows,

det V = ∑_{\begin{matrix}I⊆[k+h] \\ |I|=k \end{matrix}} ϵ(I, [k]) det V_{[k],I} det V_{[k]^{c},I^{c}} .

As the summand with index I has a factor ϵ^{∑_{j∈I^{c}}(γ_{j}+k+h−j)}, the term with I = [k] has a factor ϵ^{|β|+\binom{h}{2}}, and all other terms contain a higher power of ϵ. Thus,

(A.3)  \frac{Δ_{γ}(z⃗, ϵy⃗)}{ϵ^{|β|+\binom{h}{2}}} = det(z_{i}^{α_{j}+k+h−j})_{1≤i,j≤k} det(y_{i}^{β_{j}+k+h−(k+j)})_{1≤i,j≤h} + O(ϵ)

= ( ∏_{i=1}^{k} z_{i}^{h}) Δ_{α}(z⃗) Δ_{β}(y⃗) + O(ϵ) .

Comparing this equation for γ = ν and γ = 0 allows us to conclude. □

The bivariate homogeneous Chebyshev polynomials of the second kind are defined by

(A.4)  U_{k}(x, y) = \frac{x^{k+1} − y^{k+1}}{x − y} = x^{k} + x^{k−1}y + ⋯ + y^{k} .

Define the staircase partition μ_{n,ℓ} as the partition

(A.5)  μ_{n,ℓ} = (ℓn − ℓ, ℓn − 2ℓ, …, ℓ, 0, 0, …) .

N \{  \overbrace{ℓ} \} m ν

Figure 3. An example of partition λ(N, m, ℓ, ν) with N = 11, m = 3, ℓ = 4 and ν = (5, 2, 1).

The associated Schur function in n variables is easily evaluated by (A.1):

(A.6)  s_{μ_{n,ℓ}}(z⃗) = \frac{Δ_{μ_{n,ℓ}}(z⃗)}{Δ(z⃗)} = \frac{Δ(z_{1}^{ℓ+1}, …, z_{n}^{ℓ+1})}{Δ(z_{1}, …, z_{n})}

= ∏_{1≤i<j≤n} \frac{z_{i}^{ℓ+1} − z_{j}^{ℓ+1}}{z_{i} − z_{j}} = ∏_{1≤i<j≤n} U_{ℓ}(z_{i}, z_{j}) .

Appendix B. Properties of staircase Schur functions

Let us consider three non-negative integers N, m and ℓ, with m ≥ 1, and a partition ν of length ℓ(ν) ≤ m, with ν_{1} − ν_{m} ≥ ℓ. We define the partition λ(N, m, ℓ, ν) as follows: for 0 ≤ i < N, consider the unique way of writing N − i = am + b, with a ≥ 0 and 0 ≤ b < m (that is, a = ⌊(N − i)/m⌋ and b ≡ N − i (mod m)). Then

λ(N, m, ℓ, ν)_{N−i} = am + ν_{b} .

(See Figure 3.) We call such diagrams m-staircase diagrams, and the Schur functions s_{N,m,ℓ,ν}(z⃗) ≡ s_{λ(N,m,ℓ,ν)}(z⃗) in N variables m-staircase Schur functions. These functions generalize the (1-)staircase and 2-staircase functions defined in (1.6) and (1.3), corresponding to taking m = 1 and 2, respectively, ν_{m} = 0, and N a multiple of m (ν_{1} ≡ ℓ′ for 2-staircase functions). In this section we set q = exp(\frac{2πi}{ℓ+m}).

We say that a symmetric function in N variables f(z_{1}, …, z_{N}) satisfies the (m, ℓ)- wheel condition if, for I = {i_{1}, …, i_{m+1}} ⊆ [N] and K = {k_{1}, …, k_{m+1}} ⊆ [ℓ + m],

f(z_{1}, …, z_{N})|_{z_{ia}=q^{ka}w, a=1,...,m+1} = 0 .

Note that, as we deal with symmetric polynomials, it is not necessary to take ordered m-tuples in place of subsets. We call a specialization z_{i_{a}} = q^{k_{a}}w of the form above a “wheel hyperplane”. Proposition 3 is the 2-staircase function specialization of the following more general proposition. The proof we produce below is a minor variation of the paragraphs concerning Theorem 4 of [18].

Proposition 10. The symmetric function in N variables s_{N,m,ℓ,ν}(z⃗) satisfies the (m, ℓ)- wheel condition.

Proof. Consider the generic wheel hyperplane z_{i_{a}} = q^{k_{a}}w for i_{a} ∈ I and k_{a} ∈ K as above. Write λ = λ(N, m, ℓ, ν) for brevity. Represent s_{N,m,ℓ,ν}(z⃗) as a ratio of a shifted Vandermonde determinant by a Vandermonde determinant, Δ_{λ}/Δ, as in equation (A.1). Since, even under the specialization, the variables z_{i} are all distinct, the Vandermonde determinant in the denominator is non-singular, so it suffices to prove that the shifted Vandermonde determinant vanishes. The shifted entries of the partition are λ̃_{i} = λ_{i}+(N −i), and, writing i = N −am−b, we have λ̃_{N−am−b} = (ℓ+m)a+b+ν_{m−b}. Note in particular that

(B.1)  λ̃_{N−am−b} ≡ b + ν_{m−b} (mod ℓ + m),

independently of a. Consider the matrix V_{ij} = zλ̃_{j} ; i, such that Δ_{λ} = det V. Let V′ denote the rectangular minor of V restricted to the m + 1 rows in I, and write j = N − am − b as above. Then, because of equation (B.1),

V′_{ij} = zλ̃_{j} ; i = wλ̃_{j}q^{k_{i}(N−(ℓ+m)a_{j}−b_{j}−ν_{b_{j}})} = wλ̃_{j}q^{Nk_{i}}q^{−k_{i}(b_{j}+ν_{m−b_{j}})} .

As b + ν_{b} for b ∈ {0, …, m − 1} takes m distinct values, V′ has rank at most m, while it has m + 1 rows. This proves that det V = 0. □

Now we present a generalization of Proposition 4.

Proposition 11. For I = {i_{1}, …, i_{m}} ⊆ [N] and K = {k_{1}, …, k_{m}} ⊆ [ℓ + m], s_{N,m,ℓ,ν}(z⃗) satisfies the recursion

(B.2) s_{N,m,ℓ,ν}(z⃗)(z⃗_{\I}, q^{k_{1}}w, …, q^{k_{m}}w)

= s_{ν}(q^{k_{1}}, …, q^{k_{m}})w^{|ν|}( ∏_{j∈[N]\I} ∏_{h∈[ℓ+m]\K}(z_{j} − q^{h}w))s_{N−m,m,ℓ,ν}(z⃗_{\I}) .

Proof. From Proposition 10 it follows that, for I and K as above, s_{N,m,ℓ,ν}(z⃗) satisfies the equation

(B.3) s_{N,m,ℓ,ν}(z⃗_{\I}, q^{k_{1}}w, …, q^{k_{m}}w) = ( ∏_{j∈[N]\I} ∏_{h∈[ℓ+m]\K}(z_{j} − q^{h}w))F_{N,m,ℓ,ν}^{(K)}(z⃗_{\I}, w) ,

for some polynomial F_{N,m,ℓ,ν}^{(K)}(z⃗_{\I}, w). Rewrite the equation above in the form

Δ_{λ(N,m,ℓ,ν)}(z⃗, q^{k_{i}}w) = Δ(z⃗, q^{k_{i}}w) ∏_{j,h}(z_{j} − q^{h}w)F_{N,m,ℓ,ν}^{(K)}.

An easy computation of minimal and maximal degree in w for all the factors in this expression other than F^{(K)} shows that F_{N,m,ℓ,ν}^{(K)}(z⃗_{\I}, w) is homogeneous of degree |ν| in w. Thus,

F_{N,m,ℓ,ν}^{(K)}(z⃗_{\I}, w) ≡ w^{|ν|} lim_{v→0} \frac{F_{N,m,ℓ,ν}^{(K)}(z⃗_{\I}, v)}{v^{|ν|}},

and, in order to determine this quantity, it suffices to divide both sides of equation (B.3) by w^{|ν|} and take the limit w → 0. Using Lemma A.2, we find for the left-hand side of equation (B.3)

lim_{w→0} \frac{s_{N,m,ℓ,ν}(z⃗_{\I}, q^{k_{1}}w, …, q^{k_{m}}w)}{w^{|ν|}} = s_{ν}(q^{k_{1}}, …, q^{k_{m}})s_{N−m,m,ℓ,ν}(z⃗_{\I}) ∏_{j∉I} z_{j}^{ℓ} .

The factor ∏_{j∉I} z_{j}^{ℓ} simplifies with the the same term appearing on the right-hand side, from the limit of the product of binomials z_{j} − q^{h}w. Therefore we end up with

(B.4)  F_{N,m,ℓ,ν}^{(K)}(z⃗_{\I}, w) = w^{|ν|}s_{ν}(q^{k_{1}}, …, q^{k_{m}})s_{N−m,m,ℓ,ν}(z⃗_{\I}) .
□

For a symmetric polynomial P (z⃗) in N variables, and 1 ≤ k ≤ N, call d_{k}(P) the maximum degree of P in (any) k variables simultaneously. In what follows, when the number of variables is clear, we will use the short notations d ≡ d_{1} and D ≡ d_{N}. Recall that, for a Schur function s_{λ}(z⃗), d_{k} = λ_{1} + ⋯ + λ_{k}.

Among the staircase Schur functions considered in the propositions above, the subclass ν_{1} = ⋯ = ν_{m} = 0 (i.e., ν = ∅) has the further property of being “of minimal degree” among all symmetric functions satisfying the wheel condition, in various senses involving this set of degrees d_{k}. The following proposition describes some of the possible choices. It is a generalization to the m-staircase case of the m = 2 situation analysed in [18, Thm. 4], but, in contrast to Proposition 10, the proof technique is substantially different, as the Lagrange interpolation argument used in [18] is specific to m = 2. (With higher values, some degree counting hypothesis is not met.)

Determining the uniqueness of a function satisfying a precise set of conditions and degree bounds is often a useful tool when one wants to “prove that two (families of) functions are the same”. Although this could appear as a rare coincidence, this line of reasoning has already proven to be of value in several enumeration problems related to integrable systems, ranging from the recognition of the Izergin determinant [6], and its identification as a Schur function [16], up to the “higher-spin” cases in [18]. We report the following result, with the hope that it may be useful in generalizations of six-vertex and loop models involving simultaneously both “higher-spin” and “higher rank”, i.e., higher values of m (besides m = 2) in representations of the quantum affine algebra q-deforming 𝔰𝔩(m). Proposition 12. Let N = am + b, with a ≥ 0 and 1 ≤ b ≤ m. The symmetric polynomial in N variables s_{N,m,ℓ}(z⃗) := s_{N,m,ℓ,∅}(z⃗) has (D, d, d_{m}) = (D^{∗}, d^{∗}, d_{m}^{∗}), with

(D^{∗}, d^{∗}, d_{m}^{∗}) = (aℓ (\frac{m(a−1)}{2} + b) , aℓ, (N − m)ℓ) .

It is the unique symmetric function (up to multiplication by a scalar) satisfying the (m, ℓ)-wheel condition and any of the following degree conditions:

(a) d ≤ d^{∗} and D ≤ D^{∗};

(b) d_{m} ≤ d_{m}^{∗};

(b′) d ≤ d^{∗} and m divides N;

(c) f_{m,ℓ}(D, d) ≤ f_{m,ℓ}(D^{∗}, d^{∗}), for f_{m,ℓ}(D, d) = \frac{ℓ}{m}D + \frac{d(d+ℓ)}{2}.

Proof. Clearly (b′) is implied by (b), as d_{m}^{∗} = m d^{∗} if m divides N, and d_{m} ≤ m d for any polynomial, so it suffices to concentrate on the three cases (a), (b), and (c). Observe that a degree condition d ≤ d^{∗} alone would fail to guarantee uniqueness, since, if b < m, any s_{N,m,ℓ,ν} such that ν has at most m − b non-zero parts and ν_{1} ≤ ℓ would work.

The fact that the Schur functions above satisfy the claimed wheel condition has been already proven in Proposition 10, and the degrees are easily calculated. So we just have to prove degree minimality, and uniqueness.

If we have a = 0 (i.e., N ≤ m), for arbitrary m and ℓ, the statement is trivial because the wheel condition is empty (there are no wheel hyperplanes), and indeed s_{N,m,ℓ}(z⃗) = 1 in this case.

The case m = 1, and arbitrary N and ℓ, is also fairly simple. A polynomial in N variables P (z⃗) satisfies the (1, ℓ)-wheel condition if and only if, for all i < j and 1 ≤ k ≤ ℓ, it is divisible by z_{i} − q^{k}z_{j}. Therefore the polynomial of minimal degree satisfying the wheel condition consists of the product of these factors, and indeed coincides with s_{N,1,ℓ}(z⃗) ≡ s_{μ_{N,ℓ}}(z⃗).

The proof for generic values of m and N, and any of the degree conditions in the list, is done by a double induction on N and m, using the cases above as a basis. Let us assume the statement to be true up to the value m − 1, and, for the value m, up to N − 1 variables. Then suppose that P (z⃗) is a symmetric polynomial in N variables which satisfies the (m, ℓ)-wheel condition, and with a degree triple (D, d, d_{m}) satisfying any of the conditions. We want to show that, up to rescaling P (z⃗) by a constant factor, P (z⃗) = s_{N,m,ℓ}(z⃗).

We know from Proposition 11 that, for I and K appropriate sets (i.e., I ⊆ [N] and K ⊆ [ℓ + m], |I| = |K| = m), P (z⃗) satisfies the equation

(B.5)  P (z⃗_{\I}, q^{k_{1}}w, …, q^{k_{m}}w) = ( ∏_{\begin{matrix}j∈[N]\I \\ h∈[ℓ+m]\K \end{matrix}} (z_{j} − q^{h}w)) F_{N,m,ℓ}^{(K)}(z⃗_{\I}, w) ,

for some polynomial F_{N,m,ℓ}^{(K)}(z⃗_{\I}, w), symmetric in the N − m = (a − 1)m + b variables {z_{j}}_{j∉I}, and satisfying the (m, ℓ)-wheel condition for the remaining variables z_{j}.

Call d(F) the maximum degree of F in one variable, seen as a polynomial in variables z_{j} only, d_{w}(F) the degree as a polynomial in w, and D_{w}(F) the maximum total degree of F in z_{j}’s and w. From the degree triple of P and the binomial factors of P, it is easy to realize that

(B.6)  D_{w}(F) ≤ D(P) − (N − m)ℓ ,
(B.7)  d(F) ≤ d(P) − ℓ ,
(B.8)  d_{w}(F) ≤ d_{m}(P) − (N − m)ℓ ≤ m d(P) − (N − m)ℓ .

(The inequalities come from the fact that cancellations may occur in P from the specialization. The equation for d_{w}(F) is obtained by considering the m-tuple of variables {z_{i}}_{i∈I} in P.) Furthermore, if N ≥ 2m, d_{m}(F) is defined, and we can also infer

d_{m}(F) ≤ d_{m}(P) − mℓ .

(This equation is obtained by considering the m-tuple of variables {z_{i}}_{i∈J}, for some J of size m and disjoint from I, in P).

From the above bounds on the degree of F, and the fact that F_{N,m,ℓ}^{(K)}(z⃗_{\I}, w) must satisfy the (m, ℓ)-wheel condition for m-tuples of the N − m remaining variables z_{j}, we can prove in the various cases one of the inequalities

D_{w}(F) ≤ D^{∗}(N − m, m, ℓ) ; (B.9a)

d_{w}(F) ≤ 0 and ( d_{m}(F) ≤ d_{m}^{∗}(N − m, m, ℓ) or d(F) = 0 ); (B.9b)

f_{m,ℓ}(D_{w}(F), d(F)) ≤ f_{m,ℓ}(D^{∗}(N − m, m, ℓ), d^{∗}(N − m, m, ℓ)); (B.9c) and by induction on N we conclude that

(B.10)  F_{N,m,ℓ}^{(K)}(z⃗_{\I}, w) = c_{K} s_{N−m,m,ℓ}(z⃗_{\I}) ,

for some constant c_{K}. However, c_{K} cannot depend on K either. This is seen by specializing equations (B.5) and (B.10) to w = 0, which gives

(B.11)  P (z⃗_{\I}, 0, …, 0) = c_{K} s_{N−m,m,ℓ}(z⃗_{\I}) ∏_{j∈[N]\I} z_{j}^{ℓ} .

So, up to a multiplicative factor in P, we know that for any I and K as above, the specializations of P (z⃗) and of s_{N,m,ℓ}(z⃗) to z_{i_{a}} = q^{k_{a}}w are equal. This can be rephrased by saying that the difference R(z⃗) := s_{N,m,ℓ}(z⃗)−P (z⃗) is a symmetric polynomial satisfying the (m−1, ℓ+1)-wheel condition, and furthermore this implies easily that D(R), d(R), and d_{m}(R) are a triple of entries smaller or equal to some triple (D, d, d_{m}) satisfying (one of) the degree condition(s) under consideration (because P does this by hypothesis, and the Schur function does it explicitly, and in the difference the degree can at most decrease due to cancellations).

As all the degree conditions in our proposition are monotone (in particular, f_{m,ℓ}(D+ α, d+β) ≥ f_{m,ℓ}(D, d) if α, β ≥ 0), the quantities in the conditions, as functions of D(R), d(R), and d_{m}(R), are bounded above by the analogous quantities as functions of D^{∗}, d^{∗}, and d_{m}^{∗} (for parameters (m, ℓ)).

Making an induction hypothesis in m, these degree bounds are to be compared with the bounds for a symmetric function in N variables, satisfying the (m − 1, ℓ + 1)-wheel condition stated in the proposition. Therefore, in our range of interest m ≥ 2, a ≥ 1, write N = am + b = ã(m − 1) + b̃, with 1 ≤ b̃ ≤ m − 1. Clearly ã ≥ a. The triple (D, d, d_{m}), pertinent to the bounds on the degrees of R for the (m, ℓ) case, reads

(B.12)  D = ℓ(m\binom{a}{2} + ab);
(B.13)  d = ℓa;
(B.14)  d_{m} = ℓ(N − m) ;

while the triple pertinent to the bounds for the (m − 1, ℓ + 1) case reads

(B.15)  D′ = (ℓ + 1)((m − 1)\binom{ã}{2} + ãb̃);
(B.16)  d′ = (ℓ + 1)ã;
(B.17)  d′_{m−1} = (ℓ + 1)(N − m + 1) .

In particular, f_{m,ℓ}(D, d) = ℓ^{2}aN/m and f_{m−1,ℓ+1}(D′, d′) = (ℓ + 1)^{2}ãN/(m − 1). As we have

(B.18a)  d < d′;
(B.18b)  d_{m−1} ≤ d_{m} < d′_{m−1} ;
(B.18c)  f_{m−1,ℓ+1}(D, d) < f_{m−1,ℓ+1}(D′, d′) ;

(the last inequality comes with some algebra: the difference is

f_{m−1,ℓ+1}(D, d) − f_{m−1,ℓ+1}(D′, d′) = −\frac{(ℓ + 1)N}{m − 1} ((ℓ + 1)ã − ℓa) − \frac{ℓ(ℓ + m)}{m − 1}\binom{a + 1}{2}

and is visibly negative), for any of the conditions in our list we reach the conclusion that R(z⃗) = 0. □

Acknowledgements

Part of the statements proven in this paper have been conjectured in September 2009, when two of us (L.C. and A.S.) had the opportunity of working together, within the programme StatComb09 at the Institut H. Poincaré – Centre Émile Borel in Paris, which we thank for support.

We thank Alain Lascoux for important discussions. In particular, at a preliminary stage of this work, he suggested to us the use of Bazin(–Reiss–Picquet)’s Theorem for dealing with compound determinants. These conversations had a crucial role in the development of our proof.

The work of L.C. has been supported by the CNRS through a “Chaire d’excellence”.

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Philippe Biane, CNRS, IGM — Université Paris-Est 77454 Marne-la-Vallée Cedex 2, FRANCE

E-mail address: biane@univ-mlv.fr

Luigi Cantini, LPTM, and CNRS — Université de Cergy-Pontoise 95302 Cergy-Pontoise Cedex, FRANCE

E-mail address: luigi.cantini@u-cergy.fr

Andrea Sportiello, LIPN, and CNRS — Université Paris-Nord 93430 Villetaneuse Cedex, FRANCE

E-mail address: Andrea.Sportiello@lipn.univ-paris13.fr