Questions tagged [schur-functions]

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13
votes
1answer
1k views

Irreducibility of Schur polynomials

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=(\...
27
votes
3answers
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+...
13
votes
0answers
1k views

Generalization of Cauchy's identity

Let $ s_{\lambda} $ be the Schur function associated to the partition $ \lambda $. Cauchy's identity (as in Macdonald) states that $$ \sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = \prod_{i,j}(1-...
10
votes
3answers
1k views

A class of matrix determinants between Wronskians and Vandermondes

Update: see below Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions $...
20
votes
1answer
908 views

Bounding Schur symmetric polynomials on the unit circle

Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by \begin{equation} s_\lambda(x_1,\ldots, x_n) = a_{\lambda + \delta}(x_1, \...
12
votes
1answer
603 views

Why do we care about Schur Positivity

Some of the most important open problems in Algebraic Combinatorics concern the Schur positivity of classes of symmetric functions. Why is this an important property to have?
9
votes
1answer
368 views

Most computationally efficient Littlewood-Richardson rule

There are many, many different versions of the Littlewood-Richardson rule: the original characterization via Yamanouchi words, Remmel's version, a description via the Poirier-Reutenauer bialgebra, the ...
7
votes
2answers
422 views

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 $\lambda =(\lambda_1,\dots,\lambda_n)$ ...
6
votes
0answers
226 views

a variational problem related to weighted logarithmic capacity

Consider the following multiple contour integral: $$ \Phi_\lambda := \oint \ldots \oint \prod_{1 \le j < k \le n} (z_j^{-1} - z_k^{-1}) \prod_{j=1}^n \prod_{k=1}^n (1 - z_j x_k)^{-1} \prod_{j=1}^n ...
2
votes
1answer
206 views

Cauchy identity, with sum restricted over partitions with first part $\leq n$

The Cauchy Identity $$ \sum_{\nu}s_{\nu}(x)s_{\nu}(y) = \prod_{j,k=1}^{\infty}\frac{1}{1-x_{j}y_{k}} $$ expresses the sum over all integer partitions of the product of pairs of Schur polynomials as ...