Questions tagged [schur-functions]

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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+...
Gjergji Zaimi's user avatar
20 votes
1 answer
1k 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, \...
John Jiang's user avatar
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18 votes
0 answers
375 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_\...
Allen Knutson's user avatar
16 votes
0 answers
545 views

Identity involving Schur polynomials, binomial coefficients and contents of partition

Let $C_{\lambda,\mu}$ be the coefficients defined as $$ s_\lambda\left(\frac{x_1}{1-x_1},...,\frac{x_N}{1-x_N}\right)=\sum_{\mu\supset \lambda}C_{\lambda\mu}s_\mu(x_1,...,x_N),$$ where $s$ are the ...
Marcel's user avatar
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15 votes
0 answers
244 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, ...
Zach H's user avatar
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14 votes
2 answers
836 views

Do you know an elegant proof for this expression for a Schur function?

I know that the identity $$ s_\mu = \sum_{\mu-\lambda \text{ is a horizontal strip}} \;\sum_{\alpha\vdash|\lambda|} \frac{\chi^\lambda_\alpha}{z_\alpha} \prod_i(p_i-1)^{a_i} $$ holds. Here $\alpha=1^{...
Amritanshu Prasad's user avatar
14 votes
2 answers
2k 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 ...
darij grinberg's user avatar
13 votes
1 answer
880 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?
Apprentice Counter's user avatar
13 votes
1 answer
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=(\...
Vladimir Dotsenko's user avatar
13 votes
1 answer
389 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. ...
Will Sawin's user avatar
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13 votes
1 answer
669 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 ...
Zach H's user avatar
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13 votes
0 answers
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-...
R. Rosenbaum's user avatar
12 votes
1 answer
807 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 ...
Nicolas Ford's user avatar
  • 1,520
12 votes
1 answer
643 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 ...
Alexander Chervov's user avatar
12 votes
2 answers
356 views

Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule

The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...
Zoltan Zimboras's user avatar
11 votes
1 answer
648 views

Generating function for Schur polynomials

Consider the generating function $$ G_n(x_1,x_2,\ldots,x_n, t_1,t_2,\ldots,t_n) =\sum_{\lambda}s_{\lambda}(x_1,x_2,\ldots, x_n) t_1^{\lambda_1}t_2^{\lambda_2} \cdots t_n^{\lambda_n}, $$ where the sum ...
Leox's user avatar
  • 546
11 votes
1 answer
282 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 ...
Peter McNamara's user avatar
11 votes
3 answers
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 $...
Alex R.'s user avatar
  • 4,902
11 votes
1 answer
470 views

Is the appearance of Schur functions a coincidence?

The Schur functions are symmetric functions which appear in several different contexts: The characters of the irreducible representations for the symmetric group (under the characteristic isometry). ...
matha's user avatar
  • 173
10 votes
1 answer
314 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)|^...
Marcel's user avatar
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9 votes
1 answer
529 views

Determinant connection between Schur polynomials and power sum polynomials

Let $f_i=f_i(x_1,x_2,\ldots, x_n),i=0,1,2, \ldots $ be a family of symmetric polynomials. For the partition $\lambda=(\lambda_1,\lambda_2, \ldots, \lambda_n)$ consider the determinant $$ D_\lambda(f)...
Leox's user avatar
  • 546
9 votes
1 answer
302 views

The convex hull of Schur polynomial evaluations

Let $r\leq n$ and $d$ be positive integers. A probability vector is a vector of non-negative entries that sum to 1. For each probability vector $\lambda$ of length $n$, let $$s(\lambda)=(\dim[\pi] \...
Ben's user avatar
  • 1,010
8 votes
2 answers
523 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 $\lambda =(\lambda_1,\dots,\lambda_n)$ ...
sometempname's user avatar
8 votes
1 answer
347 views

Harmonic flow on the Young lattice

Let me begin with some preliminary concepts: A positive real-valued function $\varphi: P \rightarrow \Bbb{R}_{>0}$ on a locally finite, ranked poset $(P, \trianglelefteq)$ is harmonic if $\varphi(\...
Jeanne Scott's user avatar
  • 1,847
8 votes
0 answers
641 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 ...
Hanxiong Zhang's user avatar
7 votes
2 answers
493 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 $s_k(x_1,x_2,x_1,x_2)$ ...
7-adic's user avatar
  • 3,764
7 votes
2 answers
255 views

About the sum of rectangular power sums

Let $n \geq 1$ be an integer and consider the symmetric function $$D_n = \sum_{d|n} p_d^{n/d},$$ where $p_{d}$ are the power-sum symmetric functions. It can be checked up to $n=35$ that the symmetric ...
F. C.'s user avatar
  • 3,507
7 votes
1 answer
217 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}\...
Sasha's user avatar
  • 1,343
6 votes
1 answer
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 ...
Maurizio Monge's user avatar
6 votes
2 answers
868 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 ...
darij grinberg's user avatar
6 votes
1 answer
1k 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...
Armin Straub's user avatar
  • 1,372
6 votes
0 answers
229 views

Derivations for symmetric functions

A symmetric function is a formal power series in infinitely many variables $x_1,x_2,\dots$ invariant under the permutation of variables (as opposed to a polynomial). Let $\Lambda$ denote the algebra ...
Zach H's user avatar
  • 1,899
6 votes
0 answers
252 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 ...
user262841's user avatar
6 votes
0 answers
254 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 ...
John Jiang's user avatar
  • 4,354
5 votes
3 answers
394 views

Applying $\sum_i \partial_{x_i}$, $\sum_i x_i \partial_{x_i}$ and $\sum_i x_i^2 \partial_{x_i}$ to Schur polynomials

The operators $L_k=\sum_i x_i^k\frac{\partial}{\partial x_i}$, with integer $k$, take symmetric polynomials into symmetric polynomials. Is it known how to write the result of the application of $L_0$, ...
thedude's user avatar
  • 1,417
5 votes
2 answers
401 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 $\...
GGT's user avatar
  • 685
5 votes
1 answer
707 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 $p_1,p_2,\dots$...
JBorger's user avatar
  • 9,268
5 votes
2 answers
304 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} ...
Fred's user avatar
  • 157
5 votes
0 answers
116 views

Sum of Schur functions associated to self-conjugate partitions

The $\tau$-function $H^\circ \big(t ;\vec{x} \big)$ associated with counting simple Hurwitz numbers is the formal power series \begin{equation} (\dagger) \quad H^\circ \big(t ;\vec{x} \big) \, = \, \...
Jeanne Scott's user avatar
  • 1,847
5 votes
0 answers
394 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 ...
Zach H's user avatar
  • 1,899
4 votes
1 answer
304 views

Nonnegativity locus of Schur polynomials

Let $a_1,\ldots,a_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such ...
Tobias Fritz's user avatar
  • 5,775
4 votes
1 answer
320 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{...
GGT's user avatar
  • 685
4 votes
2 answers
451 views

About Cauchy identity for Schur polynomials

(This was originally posted here, https://math.stackexchange.com/questions/4687466/cauchy-identity-for-schur-functions, and I am reposting it here as it seems to be more appropriate.) PRELIMINARY. The ...
Giulio R's user avatar
  • 121
4 votes
1 answer
233 views

Significance of partition containment in representation theory of $\operatorname{GL}_n$

I'm working on a Schur positivity problem and I came across a series of Schur polynomials (all in $n$ variables) whose indexing partitions are (conjecturally) contained in one of some given set of ...
Tanny Libman's user avatar
4 votes
1 answer
448 views

Is this simple symmetry of Littlewood-Richardson coefficients known?

Let $\lambda$ be a partition with at most $p$ parts, let $\mu$ be a partition with at most $q$ parts, and let $\nu$ be a partition with at most $p+q$ parts. Let $m\geq \nu_1$ be an integer. We denote ...
Matt Samuel's user avatar
  • 1,978
4 votes
1 answer
237 views

proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions"

I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...
dash1729's user avatar
4 votes
1 answer
216 views

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, ...
Nadia SUSY's user avatar
4 votes
2 answers
304 views

LGV scheme for lattice paths that move in non-unit spatial positive steps

In the Lindström–Gessel–Viennot lemma (LGV) applied to the $Z^2$-lattice paths are taken to move in unit spatial-steps in unit time (see here). What do we mean by "time"? In the language of ...
Thomas Kojar's user avatar
  • 4,414
4 votes
1 answer
303 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{...
Satoshi  Nawata's user avatar
4 votes
0 answers
96 views

Representation-theoretic interpretation of double Schur polynomials

The Schur polynomials $$s_\lambda(x_1, \ldots, x_n) = \frac{|x_i^{\lambda_j+n-j}|_{1\le i,j\le n}}{|x_i^{n-j}|_{1\le i,j\le n}}$$ naturally appear as polynomial representatives for Schubert classes in ...
Antoine Labelle's user avatar