44 problems

Schur positivity conjecture. The coefficients $a^\nu_{\lambda\mu}(t)$ are nonnegative integer polynomials in $t$.

Let $\mu=(\mu_1,\ldots,\mu_m)$ be a partition and let $r\geq m$. Define … Here $r\mu=(r\mu_1,\ldots,r\mu_m)$. Monotonicity conjecture. For consecutive values of $j$, the difference…

Let $n$ and $k$ be the parameters in the Delta Conjecture, let $s_\lambda^\perp$ denote the operator adjoint to multiplication by $s_\lambda$, let $H_\lambda(x;q)$ be the Hall–Litt…

Parabolic Hall–Littlewood polynomials are the Catalan functions associated with concatenations of empty root ideals and a partition weight. Their Schur expansion has coefficients r…

Let $\mathfrak g=\mathrm D_{n+1}^{(2)}$, let $m$ be a nonnegative integer, and set $\Lambda=2m\Lambda_0$. Define … Let $V(\Lambda)$ be the integrable highest-weight representation…

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 $\bar Q'_\mu$,…

Generalized dominant-weight spanning conjecture. For all $\gamma^{(i)}\in\mathbb{Z}^{\eta_i}$, the product

Let $\lambda$ and $\mu$ be partitions, let $\widetilde{H}_{\mu}[X;0,t^{-1}]$ be the modified Hall–Littlewood polynomial specialized at $q=0$, let $\nabla$ be the nabla operator, an…

Let $\lambda$ and $\mu$ be partitions, let $\widetilde{H}_{\mu}[X;0,t]$ be the modified Hall–Littlewood polynomial obtained by specializing the modified Macdonald polynomial at…

Let $k$ be a nonnegative integer, let $x=(x_1,\dots,x_n)$, and let $\ddot{e}_r(x)$ denote the notation used in the source. Additional type B affine Jacobi–Trudi conjecture. With…

Let $k$ be a nonnegative integer, let $x=(x_1,\dots,x_n)$, and write $\dot{e}_r(x)=e_r(x^{\pm})$. For the type $\mathrm{B}_n$ Hall–Littlewood polynomial…

Let $n$ be a positive integer and let $p$ be a non-negative integer. For a partition $\lambda=(\lambda_1,\ldots,\lambda_n)$, let $m_r(\lambda)$ denote the number of parts equal to…

Let $n$ be a positive integer and let $p$ be a non-negative integer. For a partition $[200~\lambda=(\lambda_1,\ldots,\lambda_n)$, let $m_r(\lambda)$ denote the number of parts equa…

Let $cmathfrak{S}_{w}cin coperatorname{Pol}_n$ be a Schubert polynomial, and let $clangle cmathfrak{S}_{w}crangle_n^q$ denote its $q$-divided symmetrization. For a partition…

Let $cmathfrak{S}_{w}(x_1,cdots,x_n)$ be a Schubert polynomial, and let $clangle cmathfrak{S}_{w}crangle_n^q$ denote its $q$-divided symmetrization. A symmetric polynomial is Hall–…

The $Y=-1$ $h$-vector conjecture. The coefficients satisfy

Let $r=\binom{n+1}{2}$, and let $h_{n,0},\ldots,h_{n,r}$ be the nonnegative integers defined by … Set $k=\binom{n-1}{2}$. The $h$-vector conjecture. The coefficients satisfy…

Let $\eta$ be a composition of length $\ell$, let $m\geq\ell$, and let $\mathcal L_m$ be the linear span of the $m$-symmetric Schur functions whose symmetric part is nonempty. Let…

Let $Z\subset\mathbb{C}^n$ be any finite $S_n$-stable set such that $\mathbb{C}[x_1,\dots,x_n]/\operatorname{gr}\mathbf{I}(Z)$ is a graded $S_n$-module, where $\mathbf{I}(Z)$ is th…

Broer's cohomological vanishing conjecture. The higher cohomology of the associated vector bundle on the flag variety vanishes. This is described as a stronger form of the Schur-po…

Let $Psi$ be a root ideal and let $mu$ be a partition. The Catalan function $H(Psi;mu)$ is a symmetric function associated with this root ideal and partition. Chen–Haiman's conject…

Let $(Psi,mu)$ be an indexed root ideal, with $mu i ttext{Par}^k$, and define $operatorname{band}(Psi,mu)i$ as in the paper. $k$-Schur positivity conjecture. If…

Let $\mu$ be a partition, let $\sigma$ range over the fillings under consideration, and let $inv_\mu(\sigma)$ and $maj_\mu(\sigma)$ denote the associated inversion and major-index…

Refined Hall–Littlewood filling identity conjecture.

Let $\lambda$ be a $(k+1)$-core with $deg(\lambda)=n$, and let $A$ range over affine Bruhat counter-tableaux of weight $1^n$ and inner shape $\lambda$. Let $spin^k(A)$ be the stati…