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### An identity related to partitions into $n$ parts and Schur polynomials

While working with Schur polynomials I found what seems like a nice identity, and I wonder if it has a simple proof. Notation: Suppose $d,n\in\mathbb{N}$, and $\lambda =(\lambda_1,\dots,\lambda_n)$ ...
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### Bounding Schur polynomials of a particular shape

Consider Schur polynomials $s_\lambda$ with $\lambda = (2m, m, m, \ldots, m, 0)$ and $\ell(\lambda) = n$ (that is, $\lambda$ has $n$ rows). Here $m \gg n$, which, for the sake of concreteness, let's ...
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### Schur functors generalization to “Jack”, “Hall-Littlewood”, “Macdonald” functors ?

Schur functors are functors from the category of vector spaces to itself. If we take an operator $M: V->V$ and apply a Schur functor to it and then calculate trace $Tr(M^{\Lambda})$ we will get ...
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A natural question covering both this and this question would be Let $n>2$. Describe Young diagrams $\lambda$ with at most $n$ nonempty rows (or equivalently non-increasing sequences $\lambda=(\... 1answer 1k views ### Sum of products of p-th powers of roots of 1 and monomial symmetric functions Hello mathematicians, i'm looking for explicit computations of expressions like $$\sum_{\substack{0\leq i,j,k<n\\i\neq j\neq k \neq i}}\zeta_n^{ip^{k_1}+jp^{k_2}+kp^{k_3}}$$ and its ... 2answers 1k views ### Majorization and Schur Polynomials Let me first define the majorization order (or dominance order) on partitions as$\lambda \succeq \mu$iff $$\sum _{i=1}^{k}\lambda_i \geq \sum_{i=1}^{k}\mu_i$$ for all$1\le k\le l-1$and$$\lambda_1+... 1answer 543 views ### Cut-and-join equation and Schur function identity This is somewhat related to my last MO post: sum of the character of the symmetric group Let$p_n$be the$n$-th Newton symmetric function, and$s_{\nu}$be the Schur function indexed by the ... 2answers 1k views ### Sym(V ⊕ ∧² V) isomorphic to direct sum of all Schur functors of V Let$V$be a finite-dimensional$K$-vector space. Then, the symmetric power$\mathrm{Sym}\left(V\oplus \wedge^2 V\right)$is isomorphic to the direct sum of all Schur functors applied to$V$(each one ... 1answer 536 views ### Are the Schur functions the minimal basis of the ring of symmetric functions with the following properties? Let$\Lambda$denote the ring of symmetric functions in variables$x_1,x_2,\dots$and with coefficients in$\mathbf{Q}$. Then$\Lambda$is freely generated as an$\mathbf{Q}$-algebra by$p_1,p_2,\dots$... 0answers 225 views ### Analogy between canonical basis of U(n_-) and Schur functors, each under restriction .1. For any category$\mathcal C$, possibly enriched over schemes, define$Rep({\mathcal C})$to be the functor category${\mathcal C} \to {\bf Vec}$with direct sum inherited from$\bf Vec$. (If$\...
Let $A$ and $B$ be two Young tableaux, i. e. Young diagrams filled with the numbers $1$, $2$, ..., $n$ for some $n$ (not necessarily the same $n$). (They need not be semistandard.) (a) (Etingof's ...