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Questions tagged [schur-functions]

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12
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1answer
257 views

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. ...
9
votes
1answer
208 views

Integral of product of Schur functions

Schur functions are irreducible characters of the unitary group $\mathcal{U}(N)$. This implies the integration formulae $$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^...
6
votes
0answers
128 views

Macdonald's “Symmetric Functions and Hall Polynomials” Section 1.5 Example 9

I'm trying to follow Example 9 in Section 1.5 of the 2nd edition of Macdonald's book "Symmetric Functions and Hall Polynomials". I have trouble with understanding some points. Before stating my ...
3
votes
0answers
92 views

Shifted schur function and holonomic

Now let us denote by $\Lambda^{*}(n)$ the algebra of polynomials in $x_{1},\ldots,x_{n}$ that become symmetric in new variables $$ x_{i}'=x_{i}-i+c, \ i \in 1,\ldots,n.$$ Here c is a arbitrary fixed ...
13
votes
0answers
144 views

Generalization of Newton's identities to Schur functions

In some recent work, I've stumbled across the following identity for $\lambda \vdash n$: $$ n s_\lambda = \sum_{k=1}^n p_k \sum_{\mu \nearrow_k \lambda} (-1)^{\mathrm{ht}(\lambda/\mu)} s_\mu. $$ Here, ...
2
votes
1answer
83 views

Frobenius coordinate expansion of character

Let $\lambda$ be the partition of integer $d$. The Frobenius coordinate of $\lambda$ is given $$ (a_1 ,\ldots,a_{d(\lambda)}|b_1,\ldots,b_{d(\lambda)}),$$ where $d(\lambda)$ denote the diagonal of $\...
3
votes
0answers
87 views

Evaluating derivatives of Schur polynomials

Given an arbitrary partition $\lambda$ and an integer $N$ (the number of variables), is there any further way to evaluate the following derivative of the Schur polynomial? \begin{align} A &= \...
2
votes
1answer
155 views

An upper bound for a vector with given norm 1 and norm 2

Suppose $X = (x_1, \ldots , x_n)$ is given and we know that $x_i$'s are nonnegative, $\sum_{i=1}^n x_i = n$ and $\sum_{i=1}^n x_i^2 = m $. Just by this information, is it possible to find a vector ...
0
votes
0answers
94 views

Methods to get Holonomic functions

Let $a_n$ be a holonomic sequence. By definition, that means there exists a linear differential equation of finite order which annihilates $F(x)$, where $F(x):=\sum a_n x^n$. Similarly let $b_n$, $...
4
votes
1answer
204 views

Decompostion of hook schur function in terms of cauchy product of holonomic functions

Let $s_{\lambda}$ denote the schur function and $\lambda$ is the partion of an integer. The schur function written in power sum symmetric basis apper as following. $\chi$ denote the character. \begin{...
11
votes
1answer
190 views

Is there a geometric interpretation of skew Schur functions?

Consider the cohomology ring of the Grassmannian of k-planes in complex n-space. It has a standard presentation as a quotient of the ring of symmetric functions. In this presentation, the Schur ...
10
votes
1answer
366 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?
3
votes
0answers
174 views

Dimension of roots of irreducible Schur polynomial on unit circle

Let $s_\lambda(x_1,\ldots,x_n)$ be a Schur polynomial in $\mathbb{C}[x_1,\ldots,x_n]$ with $\lambda=(\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n=0)$ and $\gcd(\lambda_1+n-1,\lambda_2+n-2,\dots,\...
3
votes
1answer
242 views

Product of Schur functions

Given two sets of variables $X=\{x_1,\cdots,x_n\}$, $Y=\{y_1,\cdots,y_m\}$, and two partitions $\lambda$ and $\mu$. Is there a formula for the product of the Schur functions $s_{\lambda}(X) s_{\mu}(Y)$...
1
vote
1answer
126 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 ...
3
votes
0answers
125 views

Shifted Schur functions

Let's fix the ground field $\mathbb{C}$. In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 Andrei Okounkov and Grigori Olshanski introduce a special basis for the center $Z(\...
6
votes
1answer
139 views

Sum of the ratios of Schur functions

There are simple expressions for the sums of linear and quadratic combinations of Schur functions over all partitions (including the empty one) $$ \sum_\lambda s_\lambda(x)=\prod_{i}\frac{1}{1-x_i}\...
2
votes
0answers
199 views

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 ...
7
votes
2answers
293 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 $\lambda =(\lambda_1,\dots,\lambda_n)$ ...
2
votes
0answers
96 views

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 ...
12
votes
1answer
600 views

Plugging $1-x$ into Schur polynomials

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 ...
6
votes
0answers
180 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
0answers
184 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 ...
0
votes
1answer
192 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 ...
7
votes
2answers
300 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 $s_k(x_1,x_2,x_1,x_2)$ ...
20
votes
1answer
754 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, \...
5
votes
2answers
258 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} ...
8
votes
1answer
258 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 ...
17
votes
0answers
309 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_\...
6
votes
0answers
315 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 ...
1
vote
0answers
196 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)...
10
votes
3answers
897 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 $...
1
vote
1answer
315 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 ...
4
votes
0answers
184 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,\...
4
votes
1answer
208 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{...
1
vote
0answers
900 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 ...
1
vote
1answer
490 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 $$...
10
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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-...
2
votes
1answer
313 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)\...
11
votes
1answer
518 views

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 ...
10
votes
1answer
889 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=(\...
6
votes
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 ...
26
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+...
7
votes
1answer
598 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 ...
14
votes
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 ...
5
votes
1answer
596 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 $p_1,p_2,\dots$...
4
votes
0answers
233 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 $\...
6
votes
2answers
780 views

What is the most general “two in one row for A & in one column for B” theorem?

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 ...
6
votes
1answer
974 views

Specializations of Schur functions at consecutive integers

Given a partition λ = (λ1, λ2, ..., λn) denote with sλ the associated Schur function. There exists a nice product formula for the principal specializations: sλ...