
q,t-Kostka Polynomials


Background

Let $$\Lambda$$ denote the algebra of symmetric functions in a finite or infinite alphabet $$X = \{x_1,x_2,\ldots\}$$ with coefficients in the field of rational functions $$\mathbb{Q}(q,t)$$. Also denote by $$\Lambda_{\mathbb{Z}[q,t]}$$ the algebra of symmetric functions in $$X$$ with coefficients in $$\mathbb{Z}[q,t]$$. Inside the plethystic brackets $$[]$$, we adopt the convention that $$X$$ stands for $$x_1+x_2+\ldots$$. For $$P \in \Lambda$$, $$P[X]$$ means $$P(x_1,x_2,\ldots)$$. We will also let $$X_n$$ denote the finite alphabet $$x_1+x_2+\ldots+x_n$$.

We denote partitions by their French Ferrers diagrams, that is with rows decreasing from bottom to top. For a partition $$\mu$$ of length no greater than $$k$$, denote by $$\mu+1^k$$ the partition obtained by prepending a column of length $$k$$ to the diagram of $$\mu$$.

Definition of q,t-Kostka Polynomials

For a partition $$\lambda$$, let $$P_{\lambda}(x;q,t)$$ denote the Macdonald polynomial of shape $$\lambda$$ and let $$Q_{\lambda}(x;q,t)$$ denote its dual. Define the integral form $$J_{\lambda}(x;q,t)$$ of these Macdonald polynomials as $J_{\lambda}(x;q,t) = h_{\lambda}(q,t)P_{\lambda}(x;q,t) = h'_{\lambda}(q,t)Q_{\lambda}(x;q,t)$ with $h_{\lambda}(q,t) = \prod_{s \in \lambda}{(1-q^{a_{\lambda}(s)}t^{l_{\mu}(s)+1})}, \;\;\;\; h'_{\lambda}(q,t) = \prod_{s \in \lambda}{1-q^{a_{\lambda}(s)+1}t^{l_{\mu}(s)}}$ where, for a cell $$s \in \lambda$$, $$a_{\lambda}(s)$$ and $$l_{\lambda}(s)$$ represent respectively the arm and leg of s in $$\lambda$$, that is the number of cells of $$\lambda$$ that are respectively strictly east and north of $$s$$.

Macdonald showed that $J_{\mu}[X;q,t] = \sum_{\lambda}{S_{\lambda}[X(1-t)]K_{\lambda\mu}(q,t)}$ for coefficients $$K_{\lambda\mu}(q,t)$$ in $$\mathbb{Q}(q,t)$$. It was conjectured that these coefficients, called the $$q,t$$-Kostka polynomials, are in fact polynomials in $$\mathbb{Z}[q,t]$$.

Proof of Polynomiality

Let $H_{\mu}[X;q,t] = J_{\mu}[\frac{X}{1-t};q,t].$ We now have direct access to the $$q,t$$-Kostka coefficients because $H_{\mu}[X;q,t] = \sum_{\lambda}{S_{\lambda}[X]K_{\lambda \mu}(q,t)}.$ Also let $H_{\mu}[X;t] = Q_{\mu}[\frac{X}{1-t};t] = \sum_{\lambda}{S_{\lambda}[X]K_{\lambda\mu}(t)}.$

The following theorem is central to Garsia and Zabrocki's proof of the polynomiality of the $$q,t$$-Kostka coefficients.

Theorem: For any linear operator $$V$$ acting on $$\Lambda$$ and $$P \in \Lambda$$ set $\tilde{V}^qP[X] = V^YP[qX+(1-q)Y]|_{Y=X}$ where $$V^Y$$ is simply $$V$$ acting on polynomials in the $$Y$$ variables. This given, if $$G_k = G_k(X,t)$$ is any linear operator on $$\Lambda$$ with the property that $G_kH_{\mu}[X;t] = H_{\mu+1^k}[X;t]$ for all $$\mu$$ of length no greater than $$k$$, then $$\tilde{G_k}^q$$ has the property $\tilde{G_k}^qH_{\mu}[X;q,t] = H_{\mu+1^k}[X;q,t]$ for all $$\mu$$ of length no greater than $$k$$. In particular, the modified Macdonald polynomials $$H_{\mu}[X;q,t]$$ may be obtained from the "Rodriguez" formula: $H_{\mu}[X;q,t] = \tilde{G_{\mu'_1}}^q\tilde{G_{\mu'_2}}^q \cdots \tilde{G_{\mu'_h}}^q \textbf{1}$ where $$\mu'= (\mu'_1,\mu'_2,\ldots,\mu'_h)$$ denotes the conjugate of $$\mu$$.

Given this theorem, the polynomiality of the $$q,t$$-Kostka coefficients follows:

Define the "trivial" operator $$TG_k = TG_k(X;t)$$ by setting for the $$\{H_{\mu}[X;t]\}_{\mu}$$ basis $TG_kH_{\mu}[X;t] = \begin{cases} H_{\mu+1^k}[X;t] & \mbox{if } l(\mu) \leq k \\ 0 & \mbox{otherwise. } \end{cases}$ The Kostka-Foulkes matrix $$K(t) = \|K_{\lambda\mu}(t)\|$$ is the transition matrix between the $$\{H_{\mu}[X;t]\}_{\mu}$$ basis and the Schur functions. Since $$K(t)$$ is unitriangular with entries in $$\mathbb{Z}[t]$$, it follows that its inverse $$H(t) = K(t)^{-1}$$ has entries in $$\mathbb{Z}[t]$$. This implies that $$TG_kS_{\lambda}[X]$$ is an integral linear combination of Schur functions. Since the operator $$TG_k$$ acts integrally on the Schur basis, the desired result $$K_{\lambda\mu}(q,t) \in \mathbb{Z}[q,t]$$ is an immediate consequence of the Rodriguez formula in the above theorem with $$G = TG$$.

References

Adriano M. Garsia and Mike Zabrocki, Polynomiality of the q,t-Kostka Revisited. Algebraic Combinatorics and Computer Science, 2001, pp 473-491.

Contributors

• Roger Tian (UC Davis)