20 problems

King's stretched Kostka positivity conjecture. The coefficients of the polynomial $K_{}(k)$ are non-negative.

Let $\bar K_{\lambda\mu}(q)$ denote the restricted Kostka polynomial, namely the coefficient of $\bar s_\lambda$ in the restricted modified Hall–Littlewood function $\bar Q'_\mu$,…

Let $\mathcal{P}$ denote the set of partitions, let $\mu\in\mathcal{P}$ and $\nu\in\mathbb{Z}^n$ satisfy $l(\mu)\leq n$ and $|\mu|=|\nu|$, and let $K_{\mu\nu}$ be the corresponding…

Let $\lambda$ and $\mu$ be partitions with $|\lambda|=|\mu|$, let $K_{\lambda,\mu}(q)$ be the Kostka–Foulkes polynomial, and let $\operatorname{Tab}(\lambda,\mu)$ denote the set of…

Generalized Q-Kostka positivity conjecture. If $\bar{\mu^*}$ is a partition, then $L_{\lambda;\mu^*}(q)$ is a polynomial in $q$ with non-negative integer coefficients. These coeffi…

Let $\lambda$ and $\mu$ be partitions, let $r$ be an integer that is not a part of either partition, and let $L_{\lambda\mu}(q)$ denote the $Q$-Kostka polynomial. Growth conjecture…

Let $\lambda$ be a shifted shape and $\mu$ a content, and suppose that a statistic $d$ on marked shifted tableaux is used to express … where the sum is over marked shifted tableaux…

Let $L_{\lambda\mu}(q)$ denote the $Q$-Kostka polynomial, with $\lambda$ and $\mu$ partitions in the indexing convention of the paper. Positivity conjecture. The $Q$-Kostka polynom…

Let $G$ be a semisimple algebraic group, and let $K_{\lambda,\mu}(N)$ be the stretched Kostka quasi-polynomial associated with dominant weights $\lambda$ and $\mu$. Suppose the deg…

For partitions $\boldsymbol{\mu}$ and $\lambda$, let $c_{\boldsymbol{\mu}}^{\lambda}(q,t)$ denote the structure coefficients introduced through the product of modified Hall–Littlew…

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, with root system $\Phi$, positive roots $\Phi_+$, simple roots $\alpha_1,\ldots,\alpha_r$, and corresponding fundamental wei…

Let $r$-partitions $\boldsymbol\lambda,\boldsymbol\mu$ of $n$ index the Kostka functions $K_{\boldsymbol\lambda,\boldsymbol\mu}(t)$ and $K^-_{\boldsymbol\lambda,\boldsymbol\mu}(t)$…

Let $m$ be a natural number, and consider polynomials of the form … Here ${\lambda}$ and ${\nu}$ range over the indexing partitions and $p(1)$ is the ordinary Kostka coefficient at…

Let ${\lambda}$ and ${\nu}$ be fixed partitions, and consider the stretched Kostka polynomial … Here $K_{k{\lambda},k{\nu}}$ is the ordinary, non-skew Kostka coefficient. King–Toll…

Kostka-positivity conjecture. Every $K_{\Omega\Lambda}(q,t)$ is a polynomial in $q$ and $t$ with nonnegative integer coefficients. The paper proves this conjecture in the stable se…

Concatenation conjecture. One has

Conjugation conjecture. The map $\psi$ satisfies

Let $J_\Lambda=h^\downarrow_\Lambda P_\Lambda$ be the integral form of the Macdonald superpolynomial, expanded in the monomial basis. The integrality conjecture. The coefficients i…

Lusztig's conjecture. The Kostka polynomial equals the q-weight multiplicity:

Let $V_i$ be Kirillov–Reshetikhin–Chari-type modules for a simple Lie algebra current algebra, let $M_{\lambda;\{V_i\}}(q)$ be the graded multiplicity polynomial of $V(\lambda)$ in…