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We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonalds ``seventh variation of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL_n(F_q).
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We give two proofs of the $q,t$-symmetry of the generalized $q,t$-Catalan number $C_{vec{k}}(q,t)$ for $vec{k}=(k_1,k_2,k_3)$. One is by MacMahons partition analysis as we proposed; the other is by a direct bijection.
In this paper, we compute explicitly the $q$-dimensions of highest weight crystals modulo $q^n-1$ for a quantum group of arbitrary finite type under certain assumption, and interpret the modulo computations in terms of the cyclic sieving phenomenon. This interpretation gives an affirmative answer to the conjecture by Alexandersson and Amini. As an application, under the assumption that $lambda$ is a partition of length $<m$ and there exists a fixed point in $mathsf{SST}_m(lambda)$ under the action $mathsf{c}$ arising from the crystal structure, we show that the triple $(mathsf{SST}_m(lambda), langle mathsf{c} rangle, mathsf{s}_{lambda}(1,q,q^2, ldots, q^{m-1}))$ exhibits the cycle sieving phenomenon if and only if $lambda$ is of the form $((am)^{b})$, where either $b=1$ or $m-1$. Moreover, in this case, we give an explicit formula to compute the number of all orbits of size $d$ for each divisor $d$ of $n$.
We relate the mixed Hodge structure on the cohomology of open positroid varieties (in particular, their Betti numbers over $mathbb{C}$ and point counts over $mathbb{F}_q$) to Khovanov--Rozansky homology of associated links. We deduce that the mixed Hodge polynomials of top-dimensional open positroid varieties are given by rational $q,t$-Catalan numbers. Via the curious Lefschetz property of cluster varieties, this implies the $q,t$-symmetry and unimodality properties of rational $q,t$-Catalan numbers. We show that the $q,t$-symmetry phenomenon is a manifestation of Koszul duality for category $mathcal{O}$, and discuss relations with open Richardson varieties and extension groups of Verma modules.
Motivated by the Hankel determinant evaluation of moment sequences, we study a kind of Pfaffian analogue evaluation. We prove an LU-decomposition analogue for skew-symmetric matrices, called Pfaffian decomposition. We then apply this formula to evaluate Pfaffians related to some moment sequences of classical orthogonal polynomials. In particular we obtain a product formula for a kind of q-Catalan Pfaffians. We also establish a connection between our Pfaffian formulas and certain weighted enumeration of shifted reverse plane partitions.
This document is a companion for the Maple program : Discrete series and K-types for U(p,q) available on:http://www.math.jussieu.fr/~vergne We explain an algorithm to compute the multiplicities of an irreducible representation of U(p)x U(q) in a discrete series of U(p,q). It is based on Blattners formula. We recall the general mathematical background to compute Kostant partition functions via multidimensional residues, and we outline our algorithm. We also point out some properties of the piecewise polynomial functions describing multiplicities based on Paradans results.
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