8 problems

Coulter–Do conjecture. There exists a collection of operators

Let $k$ be a positive integer, let $\mathcal{Z}\mathcal{V}_k$ be the coset-type-constant subspace, let $\mathfrak{p}_\lambda$ be the $b$-deformed class vector for $\lambda\vdash k$…

Let $k$ be a positive integer, let $\mathcal{Z}\mathcal{V}_k$ be the subspace of vectors constant on coset-type, and let $\mathfrak{p}_\lambda,\mathfrak{w}_\lambda$ be the $b$-defo…

Let $k$ be a positive integer, let $\mathcal{J}_1,\ldots,\mathcal{J}_k$ be the $b$-deformed Jucys–Murphy operators, and let $\mathfrak{e}_k$ be the identity pair partition. Define…

McNamara's center conjecture. The center $Z(\mathscr{H}_{n,K}^\Lambda)$ is precisely the set of symmetric polynomials in $\mathcal{L}_1,\ldots,\mathcal{L}_n$.

Let $\alpha$ be fixed, let $\rho$ be a partition, and let $a_{\rho}^{k,(\alpha)}$ be the coefficients defined by … For a partition $\rho$, write $\rho\cup(m)$ for adjoining a part…

Let $F_I^{(u)}(z;|\rho|)$ denote the component of index $u$ in the expansion of $F_I(z,t;|\rho|)$, and let $p_I(z;w)$ be the associated rational function. For $u>1$, the conjecture…

Let $J_1,J_3,\dots,J_{2n-1}$ be the odd Jucys–Murphy elements, let $P_n$ be the relevant distinguished element, and let $\mathbb{S}$ denote the algebra of symmetric functions. Writ…