175 problems

Let $\mathrm{DH}_n$ denote the space of diagonal harmonics, viewed as a bigraded $S_n$-module, and let $\operatorname{Frob}(\mathrm{DH}_n)$ denote its bigraded Frobenius characteri…

Let $(\mathbf {U},\mathbf{B},\gamma)$ be a commutative triple. Let $W_\Sigma$ be the restricted Weyl group acting on the set $\mathfrak{B}(\gamma)$ of relevant $M$-types, let…

Let $J_{\lambda}(q,t)$ be the integral form Macdonald symmetric function and let $s_{\mu}$ be the Schur function, with $\lambda$ and $\mu$ partitions. For a non-negative integer…

Let $\mathcal{C}$ be a generic tuple of conjugacy classes, let $\bs\mu$ encode the eigenvalue multiplicities, and let $\mathcal{M}_{\mathcal{C}}$ be the associated character variet…

A specialization $\rho$ of the algebra of symmetric functions $\operatorname{Sym}$ is an algebra homomorphism from $\operatorname{Sym}$ to $\mathbb{C}$. It is Macdonald nonnegative…

Let $e_n$ be the elementary symmetric function, let $\Delta'_{e_{k-1}}$ be the modified Macdonald eigenoperator, and let $\mathcal{LD}_n$ be the set of labeled Dyck paths of size…

Let $\Lambda$ range over superpartitions, and let $m_\Lambda$ denote the monomial basis of symmetric superfunctions. Equip the space of symmetric superfunctions with the scalar pro…

Monical–Tokcan–Yong's conjecture. If $\beta\in\widehat{\mathcal{P}}_{\alpha}$ and $\beta\in\mathbb{Z}^n_{\geq 0}$, then $\beta\leq_{S}\alpha$.

Let $R_n^{(2,2)}$ be the corresponding four-graded coinvariant ring, let $e_r$ be the elementary symmetric function, let $\Theta_{e_r}$ be the Theta operator indexed by $e_r$, let…

Let $R_n^{(2,1)}$ denote the corresponding triply graded coinvariant ring, with Frobenius series in variables $q,t,u$, and let $e_r(\mathbf{z})$ be the elementary symmetric functio…

Monomial-product conjecture.

Let $\lambda$ and $\mu$ be partitions of $n$, let $k\geq 1$, let $\nabla$ be the nabla operator on modified Macdonald polynomials, and let $(\,, \,)$ denote the Hall inner product.…

Bergeron's conjecture. Bergeron conjectured that several such symmetric functions $G[X;q]$ have this property.

Higher-order Macdonald positivity conjecture. For every partition $\mu$, the multivariate $q,t$-Kostka number $\tilde{K}^{(q,t)}_{\mu;\lambda^1,\dots,\lambda^r}$ is a polynomial in…

Let $\lambda^1,\dots,\lambda^r$ be partitions. For $I\subseteq [r]$, let $\lambda^I$ denote the combined partition used in the Macdonald polynomial $J_{\lambda^I}^{(q,t)}$, and let…

Let $\Phi^{\vec v}_{\vec u}(w):\mathcal{F}_{\vec u}\to\mathcal{F}_{\vec v}$ be the normalized intertwining operator, and let $|K_{\vec\mu}\rangle$ and $\langle K_{\vec\lambda}|$ be…

Cherednik–Orr conjecture. The PBW character satisfies

Lapointe–Vinet's operator-identification conjecture. The creation operators satisfy

Lapointe–Vinet's creation-operator conjecture. The Macdonald polynomials are given by

Macdonald's evaluation conjecture. For general parameters, the evaluation formula

Uniqueness criterion for polynomial solutions. The equation has a polynomial solution only when the parameters $h_j$ are the values specified by $$ and $lambda$ has the displayed i…

Let $n$ be a positive integer, let $g$ and $ell$ be parameters, and let $\lambda=(\lambda_1,\ldots,\lambda_n)$. Write $(y;q)_m$ for the $q$-Pochhammer symbol and…

Hausel–Letellier–Villegas' conjecture. The mixed Hodge polynomials of character varieties of Riemann surfaces with semisimple conjugacy classes at the punctures are governed by Mac…

Let $H_\mu(x;q,t)$ be the modified Macdonald polynomial and write its expansion in the $k$-Schur basis as … with…

Three-variable basic hypergeometric conjecture. The series