Proof of some Littlewood identities conjectured by Lee, Rains and Warnaar

We prove a novel pair of Littlewood identities for Schur functions, recently conjectured by Lee, Rains and Warnaar in the Macdonald case, in which the sum is over partitions with empty 2-core. As a byproduct we obtain a new Littlewood identity in the spirit of Littlewood's original formulae.


Introduction
The classical Littlewood identities are the following three summation formulae for Schur functions: where x = (x 1 , x 2 , x 3 , . . . ) is a countable alphabet.Here and throughout the rest of the paper "λ even" means the partition λ has only even parts and λ ′ denotes the conjugate of λ.These identities were first written down together by Littlewood [16, p. 238], however (1.1a) was already known to Schur [27].They have since afforded many far-reaching generalisations and have found applications in areas such as combinatorics, representation theory and elliptic hypergeometric series.In particular there are many generalisations of (1.1) at the Schur level [3,7,10,11,12,13,21,22,28].Also see [25] for comprehensive references to the literature.
The purpose of this note is to prove the Schur function case of a pair of Littlewood identities for Macdonald polynomials recently conjectured by Lee, Rains and Warnaar [15,Conjecture 9.5].To state these we need some notation.Denote the multiset of hook lengths of a partition λ by H λ .We refine this by writing H e/o λ for the submultiset of even/odd hook lengths.The standard infinite q-shifted factorial is given by (a; q) ∞ := i 0 (1 − aq i ) and we define a statistic (1.2) ς(λ) := (i,j)∈λ in terms of the Young diagram of λ; see Subsection 2.1 below.Finally, let ΛF denote the completion of the ring of symmetric functions over the field F with respect to the natural grading by degree.
Theorem 1.1.As identities in ΛQ(q) at the alphabet x = (x 1 , x 2 , x 3 , . . . ) we have that The condition 2-core(λ) = 0 generalises both the even row and even column conditions of (1.1b) and (1.1c).Indeed, by Lemma 2.2 we have that ς(λ) = 0 if and only if λ is even.Thus when setting q = 0 (1.3) and (1.4) collapse to (1.1b) and (1.1c) respectively.In this sense these identities are in the spirit of Kawanaka's identity [13,Theorem 1.1] since this reduces to (1.1a) when q = 0. Unlike Kawanaka's identity one can make sense of the q → 1 limit of (1.3) and (1.4).In either case we obtain the following Littlewood-type identity.
Corollary 1.2.As an identity in ΛQ at the alphabet x = (x 1 , x 2 , x 3 , . . .), The outline of the paper is as follows.In the next section we give preliminaries regarding partitions, Schur functions and Koornwinder polynomials.In Section 3 we prove a pair of vanishing integrals for Schur functions again conjectured by Lee, Rains and Warnaar in the Macdonald case [15, Conjecture 9.2].Then, in Section 4, we follow the techniques of [25] to prove the bounded analogues of Theorem 1.1 conjectured in [15,Conjecture 9.4].The theorem then follows by taking an appropriate limit.We conclude with a derivation of Corollary 1.2.

2.
Partitions and (BC n )-symmetric functions 2.1.Partitions.A partition λ = (λ 1 , λ 2 , λ 3 , . . . ) is a weakly decreasing sequence of nonnegative integers such that finitely many λ i are nonzero.The sum of the entries is denoted |λ| := λ 1 + λ 2 + λ 3 + • • • and if |λ| = n we say λ is a partition of n.Nonzero entries are called parts, and the number of parts is called the length, denoted l(λ).We denote by P the set of all partitions and by P n the set of all partitions with length at most n.In particular P 0 = {0} where 0 denotes the unique partition of zero.If λ ∈ P n we write λ + δ for the partition (λ The number m i (λ) of occurrences of an integer i as a part of λ is called the multiplicity.Sometimes we express a partition in terms of its multiplicities as . A partition is identified with its Young diagram, which is the left-justified array of squares with λ i squares in row i with i increasing downward.For example is the Young diagram of (6,4,3,1).The conjugate of a partition, written λ ′ , is obtained by reflecting the Young diagram in the main diagonal, so that (6, 4, 3, 1) ′ = (4, 3, 3, 2, 1, 1).The arm and leg lengths of a square s = (i, j) ∈ λ are given by a(s) := λ i − j and l(s) := λ ′ j − i, which are the number of boxes strictly to the right and below s respectively.The hook length is the sum of these including s itself, so that h(s) := a(s) + l(s) + 1.Using the same example as above, in the Young diagram s we have labelled the square s = (2, 2) so that a(s) = 2, l(s) = 1 and h(s) = 4.As in the introduction we denote the multiset of hook lengths of λ by H λ .This is further refined as H e λ and H o λ , the multisets of hook lengths which are even or odd, respectively.In terms of these we define the hook polynomials which are invariant under conjugation of λ.For z ∈ C we also need the content polynomials In this paper we will frequently encounter partitions with empty 2-core, written 2-core(λ) = 0.One definition of such partitions is that their diagrams may be tiled by dominoes.Our running example of (6, 4, 3, 1) has empty 2-core since it admits the tiling which is clearly not unique.We will now give some conditions which are equivalent to λ having empty 2-core which all easily follow by induction on |λ|.The reader interested in more general statements involving Littlewood's decomposition of a partition into its r-core and r-quotient for all r 2 may consult, for example, [17] or [19, p. 12-15].Lemma 2.1.For λ ∈ P 2n the following are equivalent: (1) 2-core(λ) = 0. ( contains n even and n odd integers.

Auxiliary results.
Here we prove some properties of the statistic ς(λ) (1.2).Firstly, as we have already used in the introduction, we have the following characterisation of the vanishing of ς(λ).
Proof.If λ is even then ς(λ) = 0 since the number of even and odd hook lengths in each row is equal.Assume that λ is not even.Then λ has an even number of odd parts.Let λ i 1 , λ i 2 be the final two odd rows of λ.Since 2-core(λ) is empty these must be separated by an even number of even rows (possibly zero).Ignoring the rows above, the contribution to ς(λ) below and including row λ i 1 may be computed as Since the numbers λ j − j are strictly decreasing this sum is positive.The next nonzero contribution to ς(λ) will come from the pair of odd rows above in the same fashion.Thus repeating the above shows that if λ has empty 2-core and contains at least two odd rows then ς(λ) > 0.
Lemma 2.3.For λ ∈ P 2n there holds Proof.We interpret the definition of ς(λ) as a sum over the Young diagram of λ where each square has weight (−1) In the Young diagram of λ+δ place the integer (−1) λ i −i−j+1 (λ i −i) in box (i, j).Summing over i, j gives the first sum on the right of (2.1).To identify the second sum, we remove the columns with index λ j + 2n − j + 1 for 2 j 2n whose entries are (−1) , which shows the first identity.The proof of the second identity is similar.Note that by (1.2), ς(λ ′ ) may be written as We thus fill the diagram of λ + δ with integers (−1) λ i −i−j+1 (2n − j), so that removing the same columns as before now gives A simple calculation shows that for 2-core(λ) = 0, (i,j)∈λ+δ completing the proof.

Schur functions.
For completeness we give a definition of the Schur functions in terms of the classical ratio of alternants.For λ ∈ P n the Schur function is defined as and s λ (x 1 , . . ., x n ) := 0 for l(λ) > n.The set of the s λ (x 1 , . . ., x n ) indexed over P n forms a Z-basis for the ring of symmetric functions in n variables, denoted Λ n .We also use the Schur functions in countably many variables x = (x 1 , x 2 , x 3 , . . .), such as in Theorem 1.1, which may be defined by the Jacobi-Trudi determinant [19, p. 41].The set of such s λ (x) when indexed over all partitions λ form a Z-basis for the ring of symmetric functions Λ.We also require the ring Λ which is the completion of Λ with respect to the natural grading by degree [23, p. 66].
Several of the results we need below are best stated in terms of Macdonald polynomials, which are a q, t-analogue of the Schur functions [19,§VI].We simply note that the Macdonald polynomials P λ (x; q, t) are a basis for Λ Q(q,t) and reduce to the Schur functions when q = t, i.e., P λ (x; q, q) = s λ (x).
2.4.Koornwinder polynomials and integrals.The Koornwinder polynomials are a family of BC n -symmetric functions depending on six parameters first introduced by Koornwinder [14] as a multivariate analogue of the Askey-Wilson polynomials [1].Here we write x = (x 1 , . . ., x n ), x ± = (x 1 , x −1 1 , . . ., x n , x −1 n ) and for a single-variable function g(x i ) we set Below the function will be one of g(x i ) = (x i ; q) ∞ or g(x i ) = (1 − x i ).Also for the infinite q-shifted factorial we adopt the usual multiplicative notation Let W := S n ⋉ (Z/2Z) n be the group of signed permutations on n letters.
under the natural action of W on the n variables where the reflections act by x i → 1/x i .For λ ∈ P n define the orbit-sum indexed by λ as where the sum is over all elements α of the W -orbit of λ, the reflections act on sequences by α i → −α i , and The orbit-sums form a basis for the ring Λ BC n of BC n -symmetric functions.For q, t, t 0 , t This further allows one to define an inner product on Λ BC n by f, g q,t;t 0 ,t 1 ,t 2 ,t 3 := where T n is the standard n-torus and the measure T (x) is given by dT The Koornwinder polynomials are defined to be the unique BC n -symmetric functions satisfying where c λµ ∈ C(q, t, t 0 , t 1 , t 2 , t 3 ), and for which Note that µ λ denotes the extension of the usual dominance order to all partitions λ, µ ∈ P: µ λ if and only if The Koornwinder polynomials satisfy many nice properties such as the quadratic norm evaluation and evaluation symmetry [4,26].The key identity we need is [25, Equation (2.6.9)](see also [23,Corollary 7.2.1]) We will only use this for λ = 0, in which case P 0 (x; q, t) = 1.
For a basis {f λ } of Λ BC n we write [f λ ]g for the coefficient of f λ in the expansion g = λ c λ f λ where the c λ lie in some coefficient ring.The virtual Koornwinder integral of a BC n -symmetric function f is defined as K (f ; q, t; t 0 , t 1 , t 2 , t 3 ) := [K 0 (x; q, t; t 0 , t 1 , t 2 , t 3 )]f.This is extended to allow for symmetric function arguments via the homomorphism Λ 2n −→ Λ BC n for which f (x 1 , . . ., x 2n ) → f (x 1 , x −1 1 , . . ., x n , x −1 n ).Of course since K 0 = 1 the orthogonality of the Koornwinder polynomials allows us to express this as Note that the denominator has the explicit evaluation 1, 1 which is Gustafson's generalised Askey-Wilson integral [9].The virtual Koornwinder integral can be evaluated for many choices of the argument f , see [15,23,24,25].In particular, the vanishing integrals of the next section may be expressed in terms of virtual Koornwinder integrals.We need one final identity involving virtual Koornwinder integrals.To state this conveniently, let

Vanishing integrals
In this section we evaluate a pair of vanishing integrals for Schur functions conjectured by Lee, Rains and Warnaar in the Macdonald case [15,Conjecture 9.2].
For a, b, q ∈ C with |a|, |b|, |q| < 1 we define where λ is a partition with length at most 2n and the normalising factor is given by Note that in terms of virtual Koornwinder integrals this is Lee, Rains and Warnaar prove the following properties of the above integral.
Lee, Rains and Warnaar also give a conjectural Macdonald polynomial analogue of this proposition [15, Conjecture 9.2].There the generalisations of (3.1) are explicit products.Our next proposition gives the evaluation of the Pfaffians in the previous proposition, verifying the conjecture of Lee, Rains and Warnaar for q = t.Proposition 3.2.For λ with length at most 2n and 2-core(λ) = 0, (3.2) . Proof.Since the structure of the Pfaffians is similar, we focus on the second identity, and evaluate (3.1b).Fix a partition λ ∈ P 2n with empty 2-core.Define the set J ⊆ {1, . . ., 2n} as the collection of integers j for which column j has a nonzero entry in the first row, and set I := {1, . . ., 2n} \ J. Since 2-core(λ) = 0 it follows that |I| = |J| = n.The elements of I and J are labeled by i k and j k respectively, where 1 k n and ordered naturally.With this established we define the n × n matrix M with entries M k,ℓ by The Pfaffian in (3.1b) may be expressed in terms of the determinant of M .Indeed, by pushing the rows with indices in J to the right we see that Pf The determinant may be evaluated simply by applying the following generalisation of Cauchy's double alternant which may be found in [5, Example 3.1; a = 0]: .
We apply this with (b, c, x k , y ℓ ) → (−1, 1, After some elementary manipulations the evaluation may now be expressed as The terms of the form 1 − q x can be simplified thanks to the identity [19, p. 10-11] where l(λ) n.Restricting all products to even/odd exponents implies that (1 − q 2i ) 2n−2i .

Bounded Littlewood identities
Here we use the integral evaluations of the previous section to prove a bounded analogue of Theorem 1.1.This is followed by proofs of the theorem and of Corollary 1.2.

4.1.
A bounded analogue of Theorem 1.1.Bounded Littlewood identities are generalisations of ordinary Littlewood identities in which the largest part of the indexing partition has an upper bound, say m, such that sending m to infinity recovers an ordinary (unbounded) Littlewood identity.The first example of such an identity was discovered by Macdonald [18,§1.5]where he used a bounded analogue of (1.1a) to prove the MacMahon and Bender-Knuth conjectures on plane partitions [2,20].Bounded analogues of the remaining two classical identities (1.1b) and (1.1c) were obtained by Désarménien, Proctor and Stembridge [7,22,28] and Okada [21] respectively.A host of other bounded identities for Hall-Littlewood and Macdonald polynomials may be found in [25] and references therein.For further discussion of the history of bounded Littlewood identities see [10].We now state the bounded analogue of Theorem 1.1.
x; q, q; q 1/2 , −q 1/2 , q 1/2 , −q 1/2 , and These identities are indeed bounded since C e λ (q −2m ; q) vanishes if λ 1 > 2m.Since, by [15,Lemma 4.1], the Koornwinder polynomials on the right reduce to classical group characters for q = 0, one recovers the previously mentioned Désarménien-Proctor-Stembridge and Okada identities respectively in this case.The Koornwinder polynomials for q = t on the right-hand side may alternatively be expressed as a ratio of determinants of Askey-Wilson polynomials [1]; see, e.g., [6,Definition 4.1].This, however, does not seem to shed light on a more explicit expression for the evaluation of these sums.In particular, the specialisations of K (m n ) above are not contained in [15,Lemma 4.1].
The following argument is sketched in [15, §9], but we give the details in the Schur case.Assuming the Macdonald polynomial version of the vanishing integrals [15, Conjecture 9.2], the same argument gives the conjectural Littlewood identities.
The q → 1 limit of the product-side of (1.4) gives the same result.The limit of either sum follows from the characterisation of partitions with empty 2-core in Lemma 2.1, namely that |H e λ | = |H o λ |.