23 problems

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…

Let $S_n$ be the symmetric group, let $0<k<n$ be an integer, let $v in Sn$ avoid the pattern $12cdots(k+1)$, and let $operatorname{Imm}v X$ denote the corresponding Kazhdan–Lusztig…

Bürgisser's conjecture. If the width of $\lambda^{(n)}$ is $\Omega(n^\delta)$ for some $\delta>0$, then the family $(\mathrm{IM}_{\lambda^{(n)}})$ is $\mathrm{VNP}$-complete.

Wang–Zhang–Zhang's immanant conjectures.

Let $k$ be a positive integer, let $\lambda$ be a partition of rank $k$, and let $G_\lambda$ be the associated Giambelli matrix. For any partition $\nu$ of $k$, let…

Let $\lambda/\mu$ be a border strip, let $\theta$ be a partition of the same size, and let $E_{\lambda/\mu}^{\theta}$ be the polynomial associated to the Jacobi–Trudi matrix…

Let $u=( u1, u2,ldots, ut)$ be a partition, and let $uvdash n$. For a skew partition $lambda/mu$ of length $n$, let $H(lambda,mu)$ denote its Jacobi–Trudi matrix, and let…

Let $\lambda$, $\mu$, $\nu$, and $\rho$ be partitions of length at most $n$, with $\mu\subseteq\lambda$ and $\rho\subseteq\nu$. Let $\operatorname{JT}_{\lambda/\mu}(\mathbf{x})$ an…

Let $mu$ and $nu$ be partitions with at most $n$ parts, and let $H{ mu/nu}$ be the Jacobi–Trudi matrix with entries $h{ mui-nuj-i+j}$. For a partition $lambda$ of $n$, let…

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 $x$ be an $n\times n$ matrix and let an $\mathfrak{S}_n$-trace immanant be an immanant $\mathrm{Imm}_{\theta}(x)$ with $\theta\in\mathcal{T}_n$. Let…

Let $x$ be an $n\times n$ matrix, let $\mathcal{T}_n$ denote the cone of $\mathfrak{S}_n$-trace immanants that are totally nonnegative (TNN), and let…

Let $\lambda$ and $(n)$ be partitions of $n$, and let $\chi$ and $\chi'$ be the corresponding irreducible characters of $\mathcal{S}_n$. Define $\lambda\preccurlyeq\mu$ to mean tha…

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 $H_{I,J}$…

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 $T(\Omega)\in\mathfrak{su}(n)$ and let $\bar T(\Omega)$ denote its relevant matrix representation. For a partition or Young diagram $\lambda$, write…

Let $T(\Omega)$ be a unitary matrix, let $\bar T(\Omega)$ denote its complex conjugate, let $\lambda$ index an irreducible representation, and let $kq$ specify a submatrix of…

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…

For a positive integer $k$, the fermionant $\mathrm{Ferm}_k$ is the matrix function obtained by weighting permutations by $k^{c(\pi)}$, where $c(\pi)$ is the number of cycles of th…

Merris–Watkins monotonicity conjecture. If $\boldsymbol{\text{\eta}}\succ\boldsymbol{\text{\eta}}'$, then