13 problems

The branching coefficient formula. The coefficients are

Let $m_0(\lambda)$ be the multiplicity of zero parts of the partition $\lambda$, let $P_\lambda(x_1,\dots,x_n;t)$ be the Hall–Littlewood polynomial, and let $b_\lambda(t)$ be its s…

Consider the classical Koornwinder–van Diejen Hamiltonians with a general one-body potential determined by a function ${\cal W}$, and let … where $L$ is the $BC_N$ Lax matrix and…

Let $K^{(n)}_\lambda(\dots z_i\dots;t_0,t_1,t_2,t_3;q,t)$ be the $BC_n$-symmetric Koornwinder polynomial, with partitions $\lambda$ and $\mu$, and define its principal specializati…

Let $\lambda$ be a partition, let $\tilde K_\lambda$ be the lifted Koornwinder symmetric function, and let $[2p_{k/2}]$ denote the specialization sending $p_{2k+1}$ to $0$ and…

Let $\lambda$ be a partition, let $\tilde K_\lambda$ be the lifted Koornwinder symmetric function, and let $[2p_{k/2}]$ denote the specialization sending $p_{2k+1}$ to $0$ and…

Let $\operatorname{Par}_n$ denote the set of partitions with at most $n$ parts, let $\lambda'\!$ be the conjugate partition, and let $N^Z_{\lambda}$ be the dual coefficient associa…

Let $\operatorname{Par}_n$ denote the set of partitions with at most $n$ parts, and let $N^{Z}_{\lambda}(\xi;\alpha,\beta,\gamma,\delta;q)$ be the coefficient associated with the r…

Let $\operatorname{Par}_n$ denote the set of partitions with at most $n$ parts. A $(0,1)$-substitution map is an operator $\varphi$ on…

Bounded Littlewood identities. Both displayed identities hold.

The Koornwinder polynomials at $q=t$ have multivariate moments indexed by a partition $\lambda$. General positivity conjecture. The positivity established for a specific $\lambda$…

The partially asymmetric simple exclusion process is described using a deformed Markov matrix, with deformation parameter $\xi$ and system variables $\mathbf{x}$. For each positive…

Let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_m)$ be a partition, and let $K_{\lambda}(\xi)$ denote the Koornwinder moment defined by … where $Z_N$ is the ASEP partition function…