# Questions tagged [schur-functions]

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### What is an orthogonal form?

I am reading the article: Algebro - Geometric applications of Schur S- and Q-polynomials On page 179, they said about an orthogonal form $\psi$ on $W$. There is no definition of an orthogonal form on ...
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### Schur polynomials with zeros in an infinite geometric progression

Let $a_1,...,a_n\in \overline{\mathbb{Q}}^\times$ be algebraic numbers, such that $\frac{a_i}{a_j}$ is not a root of unity for $i\neq j$. Furthermore, let $m\in\mathbb{N}$, $m>1$. Question: Is ...
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### Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)

For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...
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### Restricted Cauchy identity

Is there some reference for sums like: $$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)t^{|\nu|}$$ $$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)\cdot|\nu|$$ (summation ...
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### Decomposing tensor powers of the fundamental representation of exceptional Lie algebras

For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, ...
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### Is there a Giambelli identity with dual representations?

For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition. ...
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### Proofs that the Plücker relations generate the ideal of the Grassmannian

Some context: The $(k,n)$-Grassmannian is the set of $k$-dimensional subspaces of an $n$-dimensional vector space $V$. It can be realized as a projective variety via the Plücker embedding, and the set ...
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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 ...
I have a symmetric Laurent polynomial $f$ in $k$ variables expressed as a linear combination of Schur polynomials. I'd like to know what happens when I make the substitution $p(x_1,\ldots,x_k)\mapsto ... 0answers 208 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 ... 0answers 230 views ### what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions. I am trying to understand first how one can define the plethysm say s_\lambda \circ s_\mu as a module in the regular representation of the symmetric group. 1)How is it connected to the plethysms ... 1answer 199 views ### Reference Request: Irreducibles of the regular representation of the permutation group is absolutely irreducible I am writing a paper(physics) where I am using the fact that the irreducible's of the regular representations of the permutation group are absolutely irreducible in the following sense. If V is an ... 2answers 352 views ### Schur polynomial, change of variable 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) ... 1answer 842 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, \... 2answers 268 views ### Dickson/determinant type polynomial (updated) For 2\leq \ell \leq k, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation} ... 1answer 316 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 ... 0answers 321 views ### Deforming a basis of a polynomial ring The ring Symm of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a \mathbb Z-basis of Schur functions \{S_\... 0answers 338 views ### Staircase Schur functions squared Let \Delta_n be the staircase-shaped partition (n-1,n-2,\dots,1). Are there any non-obvious combinatorial objects that index s_{\Delta_n}^2? Here, s_\lambda is the Schur function indexed by ... 0answers 203 views ### Explicit basis/weight vectors for irreducibles inside the plethysm Sym^m(\bigwedge^p \mathbf(V)) This is a follow up to this question about finding the multiplicities of irreducible representations restricted to Young diagrams of 2-columns or less, inside the plethysm Sym^m(\bigwedge^p \mathbf(V)... 3answers 944 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 ... 1answer 369 views ### A Simple Bijective Proof Of Stanley's Hook-Content Formula for Hook Shapes this is my first post on math overflow so I hope it goes well. I believe I have a fairly simple bijective proof for Stanley's Hook-Content Formula in the case of hook shapes. I wanted to see if ... 0answers 186 views ### Optimization problem involving Multivariate Normal I use \phi(t) to describe the standard normal distribution density and \Phi(t) as the normal distribution CDF and would like to prove that for all n\geq3, the function:$$h(\mu_{1},\ldots,\... 1answer 229 views ### Generalization of Frobenius formula involving Macdonald polynomials Given a vector$\vec k=(k_1,k_2,\cdots)$with$k_i$are non-negative integers, the Newton polynomial$p_{\vec k}(x)$is defined as \begin{equation} p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~, \end{... 0answers 1k views ### Every antisymmetric (alternating) polynomial is divisible by Vandermonde product I am looking for a proof of the statement: Every antisymmetric (alternating) polynomial is divisible by Vandermonde product, yielding a symmetric polynomial in the result. The statement is really ... 1answer 581 views ### principal specialization of projective Schur functions Is there a nice/known formula for the (some) principal specialization of a projective (P or Q) Schur polynomial? That is, I am talking about a projective analogue of Stanley's hook content formula $$... 0answers 1k views ### Generalization of Cauchy's identity Let s_{\lambda} be the Schur function associated to the partition \lambda . Cauchy's identity (as in Macdonald) states that$$ \sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = \prod_{i,j}(1-... 1answer 351 views ### Schur polynomials in the Chern classes as direct images Let$E\to X$be a rank$r$holomorphic vector bundle on a$n$-dimensional compact complex manifold. Then, it is well known that one can recover the Segre classes of$E$as follows. Let$\pi\colon P(E)\...
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 ...