All Questions

Given the famous Littlewood-Richardson rule, in terms of Schur polynomials: $$s_\mu s_\nu=\sum_\lambda c^{\lambda}_{\mu\nu} s_\lambda,$$ is there a classification of the cases where the LR ...

There is a way to characterize for which $x_1,...,x_d$ a Schur polynomial, that can be defined as $$s_\lambda(x_1,...,x_d)=\sum_{T\in SSYT(\lambda)}x_1^{t_1}...x_d^{t_d}, $$ with the sum running over ...

Let $K$ be a field of characteristic 0 (maybe it works for more general fields) and $K[x_1,...,x_n]$ the polynomial ring in $n$ variables. Let $e_1,e_2,...,e_n$ denote the elementary symmetric ...

Consider the Hall-Littlewood polynomial $$ P_\lambda(x_1,\ldots,x_n;t)=\sum_{\sigma\in S_n/S_n^\lambda}\sigma\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod\limits_{\lambda_i>\lambda_j}\dfrac{x_i-...

Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$. Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ ...

Think of a Young diagram as a collection of rows with numbers of elements $\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...