Changing parameters in Hall-Littlewood functions

Introduction

The Hall–Littlewood $P$-functions $P_\lambda(u)$ are a one-parameter family of symmetric functions depending on a parameter $u$ and indexed by integer partitions $\lambda$. These functions interpolate between two important bases of the space of symmetric functions: the Schur functions $s_\lambda= P_\lambda(0)$ and the monomial symmetric functions $m_\lambda= P_\lambda(1)$.

The transition matrix coefficients expressing the Schur functions in terms of the monomial symmetric functions are the Kostka numbers $K_{\lambda,\mu}$,

This equation is a direct shadow of the weight space decomposition of the irreducible $GL_n(\mathbb{C})$ representations (for large enough $n$), and Kostka numbers are the dimensions of the weight spaces of irreducible $GL_n(\mathbb{C})$-representations. There are many combinatorial models which enumerate these Kostka numbers, one of the most popular being the semistandard Young tableaux: $K_{\lambda,\mu}$ is the number of semistandard Young tableaux of shape $\lambda$ and content $\mu$.

A $u$-analogue of the above picture is obtained by considering the transition matrix coefficients from the Schur functions expanded in the Hall–Littlewood $P$-basis; the coefficients $K_{\lambda,\mu}(u)$ are called the Kostka–Foulkes polynomials.

Under the Satake isomorphism this equation is a shadow of the change of basis between the canonical and standard bases of the spherical Hecke algebra, and the polynomials $K_{\lambda,\mu}(u)$ are certain Kazhdan–Lusztig polynomials, encoding the Poincaré polynomials of certain intersection cohomology. From this standpoint $K_{\lambda,\mu}(u) \in\mathbb{Z}_{\ge0}[u]$, polynomials with positive integral coefficients. Lascoux and Schützenberger developed a subtle statistic on semistandard Young tableaux, called charge and denoted $\operatorname{ch}$ in this paper, which serves to give a combinatorial interpretation of these coefficients:

where $\operatorname{SSYT}(\lambda,\mu)$ denotes the set of semistandard Young tableaux of shape $\lambda$ and content $\mu$.

In another direction, one can generalize (1) by changing the Schur function side to Hall–Littlewood $P$-functions; in this case Macdonald [13] has an expansion formula by summing over semistandard Young tableaux again:

where $\psi_T(u) \in\mathbb{Z}[u]$ is explicitly described in § 2.8.

The transition coefficients $b(q,t)$

In view of the above it is desirable to study the transition matrix between two families of Hall–Littlewood $P$-functions, parametrized by different parameters. More specifically, consider $\Lambda= \Lambda_{\mathbb{Q}(q,t)}$, the space of symmetric functions with coefficients from $\mathbb{Q}(q,t)$. The families $(P_{\lambda}(t))_{\lambda\in\operatorname{Par}}$ and $(P_{\lambda}(q))_{\lambda\in\operatorname{Par}}$ are both $\mathbb{Q}(q,t)$-bases of $\Lambda$, where $\operatorname{Par}$ denotes the set of integer partitions. We want to investigate properties of $b_{\lambda,\mu}(q,t)$ defined by

Writing the matrices $b(q,t) = (b_{\lambda,\mu}(q,t))_{\lambda,\mu\in\operatorname{Par}}$, $K(u) = (K_{\lambda,\mu}(u))_{\lambda,\mu\in\operatorname{Par}}$, we get

Carbonara [7] obtained a combinatorial formula for the entries of the inverse Kostka–Foulkes matrix, thus in principle we have a signed formula for $b_{\lambda,\mu}(q,t)$. By the above discussion

A tableaux formula for hook shapes

It would be very interesting if one could find a formula that attaches a weight to tableaux in such a way that at $t = 0$ and $q = 1$ the weights specialize to the charge and Macdonald’s monomial expansion weights term-by-term, respectively. We answer this question in the positive for the case when $\lambda$ is a hook shape and some more simple cases. We now describe our result for hook shape.

We defer the definition of charge to § 2.10; for now we only comment that given a tableau $T \in\operatorname{SSYT}(\lambda,\mu)$, for some partitions $\lambda,\mu$, each cell $x$ in the Young diagram of $\lambda$ gets assigned a weight $\operatorname{ch}_T(x)$ such that $\operatorname{ch}(T) = \sum_{x\in\lambda}\operatorname{ch}_T(x)$.

Suppose $\lambda=(n-k,1^k)$, $n-k>1$, $k\geq0$. For $T\in\operatorname{SSYT}(\lambda,\mu)$ we define $\psi_T(q,t)\in\mathbb{Z}[q,t]$ below. Let

Define

where $\operatorname{Left}(x)$ denotes the box immediately on the left of $x$ in the same row. Let

where $[m]_t=1+t+\cdots+t^{m-1}$ for $m\in\mathbb{Z}_{>0}$. Then

and $\psi_T(1,t)=\psi_T(t)$.

For the unique tableau of shape $(1^n)$ let $\psi_T(q,t)=1$. Then we prove the following theorem.

The proof of Theorem 1.1 is given in § 6. When $\lambda$ is a single row, (10) simplifies to the following product formula (proved in Theorem 5.1):

where $\ell=\ell(\mu)$ and $n(\mu)=\sum_{i\geq1}(i-1)\mu_i$.

In the final two sections § 8 and § 9 we show that analogous formulas work when $\mu$ is of the form $(n,k,1)$ and small hooks $(n-2,1^2),(n-3,1^3)$.

A positivity conjecture and further results

Theorem 1.1 immediately implies that in the case that $\lambda$ is of hook shape, we have $b_{\lambda,\mu}(t+1,t)\in\mathbb{Z}_{\geq0}[t]$. We conjecture that this is true for general $\lambda,\mu$, and have checked it for partitions of size at most 26 in SageMath.