11 problems

Let $n$ be a positive integer, let $\lambda\vdash n$ be a partition, and let $\phi^\lambda$ denote the corresponding monomial trace. For an $n\times n$ matrix $x$, write…

Wang–Zhang–Zhang's immanant conjectures.

Let $A$ be a positive semidefinite Hermitian matrix, let $\lambda\vdash n$ be a partition, let $\chi^{\lambda}$ be the irreducible character of $\mathfrak{S}_n$ indexed by…

Let $A$) be a totally nonnegative $n\times n$ matrix, meaning that all its minors are nonnegative, and let $\lambda\vdash n$ be a partition. For a class function $\phi^{\lambda}$ o…

Stanley–Stembridge decomposition conjecture. Every immanant character $\Gamma^\theta_{\mu/\nu}$ is a non-negative integral sum of Stanley–Stembridge characters.

Stembridge's conjecture. Monomial immanants of Jacobi–Trudi matrices are Schur-positive.

Let $H=(h_{i,j})_{i,j\geq 0}$ be the Hankel matrix defined in the paper, and for $k\geq 1$ let $I=(i_1,\ldots,i_k)$ and $J=(j_1,\ldots,j_k)$ be sequences satisfying … Let $H_{I,J}$…

Goulden–Jackson conjecture. For every $\theta\in\Theta$, the value of the linear extension of $\chi^{\lambda}$ to the group algebra satisfies

Let $0<k<n\leq m$ be integers, let $v\in S_n$ avoid the pattern $12\cdots(k+1)$, and let $R,C\in\binom{[m]}{n}$, where repetitions are allowed. Strong pattern-avoidance conjecture.…

Let $\operatorname{Imm}_v X$ be a Kazhdan–Lusztig immanant that is $k$-positive, and let $R,C$ be index sets for which the dual canonical basis element…

Let $\lambda(n)$ be any family of Young diagrams, where the depth of $\lambda(n)$ is the number of rows in the diagram. Assume that, for some constant $\delta>0$, the depth is…