All Questions
Tagged with rt.representation-theory symmetric-functions
99 questions
3
votes
0
answers
73
views
While expanding Jack polynomials in monomial basis
Denote $\mathbf{z}=(z_1,\dots,z_n)$. Let $P_{\kappa}(\mathbf{z};\alpha)$ be the symmetric Jack polynomials and suppose they are expanded in terms of the monomial symmetric basis $m_{\rho}(\mathbf{z})$ ...
- 43.2k
8
votes
0
answers
145
views
Asymptotics of generalized exponents of highest weight modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $H^k$ be the space of homogeneous degree $k$ harmonic polynomials in $\mathrm{Sym}(\mathfrak{g}^*)$ and $H\subset\mathrm{Sym}(\mathfrak{g}^*)...
- 413
2
votes
0
answers
57
views
Hall-Littlewood polynomials for $n$-tuples that are not partitions
For some calculations related to the unramified principal series of ${\rm GL}(n)$ over a $p$-adic field, I need to compute Hall-Littlewood polynomials that are associated to $n$-tuples that are not ...
- 6,349
1
vote
1
answer
95
views
Representation of equivariant maps
Let $n,m,k$ be positive integers. Consider the action of symmetric group $S^n$ on $\mathbb{R}^{n\times i}$ (for $i\in \{m,k\}$) by permuting rows; i.e. for each $\pi\in S^n$ and every $n\times i$ ...
- 5,405
7
votes
1
answer
298
views
Jacobi-Trudi-like identity with dual characters
If $\lambda$ is a partition with at most $n$ parts, let $s_\lambda$ be the corresponding Schur polynomial in $n$ variables $x_1,\ldots,x_n$. In particular, for $a \geq 0$, $s_{a}$ is the complete ...
- 980
5
votes
0
answers
167
views
Bounding elementary symmetric polynomials away from zero
Let $2 \leq m \leq n$ be integers and let $\mathbf{x} \in \mathbb{R}^n$ (importantly, I am not assuming that the entries of $\mathbf{x}$ are non-negative). The elementary symmetric polynomials are ...
- 6,351
0
votes
0
answers
77
views
Generating function for dimensions of the space of polynomials fixed by a single permutation
Consider the space of polynomials with complex coefficients $\mathbb{C}[x_1,x_2,\dots,x_n]$ and let $\sigma$ be a permutation of $\{1,2,\cdots, n\}$ that acts on
this space via $\sigma(x_i)=x_{\sigma(...
- 141
7
votes
0
answers
132
views
Relation between Fourier series and Schur polynomials
Asked initially at MSE.
I would like to know how to express the Fourier series of a symmetric function, $f(\theta_1,...,\theta_N)$, in terms of Schur polynomials $s_\lambda(x_1,...,x_N)$ in the ...
- 1,549
3
votes
0
answers
72
views
How to multiply dots with Young idempotents in the degenerate affine Hecke algebra (type A)
Let $\widehat{\cal H}_n$ be the type A degenerate affine Hecke algebra on $n$ strands, and let $x_1,\cdots,x_n$ be the dots. Inside of this algebra lies the algebra $\mathbb C S_n$, and the Young ...
- 311
3
votes
1
answer
107
views
Does the reproducing property of the unitary group Poisson kernel require a multiple of the identity?
The Poisson kernel of the unitary group is
$$ P(Z,U)=\frac{\det(1-ZZ^\dagger)^N}{\det(1-ZU^\dagger)^N\det(1-UZ^\dagger)^N}.$$
It has a reproducing property, $\int dU P(Z,U)f(U)=f(Z)$, where $dU$ is ...
- 2,552
1
vote
0
answers
151
views
Efficient decomposition algorithm for characters of symmetric groups
Let $\chi$ be a rational character of $G:=S_n$, and we want to know whether it decomposes into irreducibles $\chi_\lambda$, for $\lambda\in\Lambda$, with $\Lambda$ given, as
$$
\chi=\sum_{\lambda\in\...
- 14k
14
votes
1
answer
660
views
Is this generalized version of plethysm Schur positive?
Question: Suppose that $f(x_1, x_2, \dots x_n)$ is a polynomial with nonnegative integer coefficients. For each permutation $\sigma\in S_n$, let $f_{\sigma}$ denote $f(x_{\sigma(1)}, \dots, x_{\sigma(...
- 85.6k
12
votes
1
answer
642
views
Is the appearance of Schur functions a coincidence?
The Schur functions are symmetric functions which appear in several different contexts:
The characters of the irreducible representations for the symmetric group (under the characteristic isometry).
...
- 193
7
votes
0
answers
176
views
The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial
I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...
- 131
5
votes
0
answers
120
views
Representation-theoretic interpretation of double Schur polynomials
The Schur polynomials
$$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 ...
- 3,446
7
votes
1
answer
354
views
Decomposition of a tensor product of representations of $\mathrm{GL}_l(\mathbb{C})$ and decomposition of Littlewood-Richardson numbers?
For a positive integer $m$, denote $T(m)=\{(\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\lambda_1\ge \lambda_2\ge\dots \ge\lambda_m\}$ and $T^+(m)=\{ (\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\...
- 960
4
votes
1
answer
185
views
Frobenius series for the $S_n$-module $\mathbb{Q}[X]$
I'm reposting this question, by recommendation of a moderator.
I'm reading Haiman's article titled Conjectures on the quotient ring by diagonal invariants. In what follows, all vector spaces and ...
- 141
8
votes
0
answers
236
views
Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule
$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...
- 241
9
votes
1
answer
216
views
Asymptotic character theory of unitary groups via shifted Schur functions
In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 by Andrei Okounkov and Grigori Olshanski it is said that one of the motivations for that paper was the asymptotic ...
- 379
9
votes
2
answers
1k
views
Using Schur-Weyl duality
I am trying to gain a better understanding of Schur-Weyl duality specifically applied to symmetric functions. My motivating example is trying to understand the Frobenius character of the multilinear ...
- 93
6
votes
1
answer
186
views
Action of $\widehat{\mathfrak{sl}_2}$ on symmetric functions with $\mathbb{Z}_{(2)}$ coefficients
It is known that there is a representation of the affine Lie algebra $\widehat{\mathfrak{sl}_q}$ (over $\mathbb{Z}$) on the algebra of symmetric functions, where the action of the Chevalley generators ...
- 3,446
8
votes
1
answer
371
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(\...
- 2,137
8
votes
0
answers
240
views
Scalar products on symmetric functions behaving like the Macdonald scalar product
The Macdonald symmetric functions (or Macdonald polynomials)
$P_\lambda(x)$ are orthogonal with respect to the Macdonald scalar
product
$$ \langle p_\lambda,p_\mu\rangle =
\delta_{\lambda\mu}z_\...
- 50.8k
6
votes
0
answers
201
views
Hall-Littlewood polynomials of non-dominant weights
$\DeclareMathOperator\SL{SL}$Let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a sequence of positive integers and let
$$
R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
- 528
6
votes
0
answers
246
views
What do Macdonald polynomials hint about $\operatorname{Rep}(S_\infty)$?
$\DeclareMathOperator\Rep{Rep}$It is well-known that the Macdonald "$P$" polynomials deform the
Jack "$J$" polynomials [1]. The latter have profound relations with
representation ...
- 5,230
2
votes
0
answers
77
views
Chapter 2 Section 2 in Macdonald's Symmetric Functions and Hall Polynomials
Throughout this post $R$ denotes a discrete valuation ring with residue field $R/\frak{m}$ being finite. I'm reading Macdonald's exposition on the Hall Algebra on page 183 and trying to make sense of ...
3
votes
0
answers
165
views
Multiplicities of irreducible $U(n)$-modules in the tensor product $V_{\lambda}\otimes V_{\mu}$
It is known that every irreducible representation of the unitary group $U(n)$ can be uniquely described by the non-increasing sequence $\lambda=(\lambda_1,\ldots,\lambda_n)$ of integers (denote the ...
- 379
3
votes
1
answer
185
views
Is there a Jacobi–Trudi formula for skew zonal polynomials?
Skew Schur polynomials are defined as $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu\nu}s_\nu$, where the Littlewood–Richardson coefficients $c^\lambda_{\mu\nu}$ satisfy $s_\mu(x)s_\nu(x)=\sum_\lambda c^\...
- 2,552
2
votes
1
answer
212
views
Changing $S_2 \wr S_n$ for $S_n \wr S_2$ in the theory of zonal polynomials
The permutation group $S_{2n}$ has $H_{2,n}=S_2\wr S_n$ as a subgroup. The plethysm $h_n(h_2)=\sum_{\lambda\vdash n}s_{2\lambda}$ is well known.
The zonal spherical functions $\omega_\lambda(g)=\frac{...
- 1,549
1
vote
1
answer
84
views
Are top Brauer characters bounded?
Let $p_\lambda$ be power sum symmetric functions. Let $s_\lambda$ and $o_\lambda$ be irreducible characters of the unitary and orthogonal groups $U(N)$ and $O(N)$, respectively (the $s$ are the Schur ...
- 1,549
5
votes
0
answers
103
views
Form on symmetric functions and their q,t- analogues
[Notations are as in Macdonald's Symmetric Functions and Hall Polynomials]
The space of symmetric functions $\Lambda_{\mathbb{Q}}$ has a bilinear form defined by $ (p_\lambda, p_\mu)= z_\lambda \...
- 820
8
votes
2
answers
564
views
"Kronecker Product" for quasi-symmetric functions
Recall that the Kronecker product
$s_\lambda * s_\mu$ of two Schur functions $s_\lambda$ and $s_\mu$ is the symmetric function
whose expansion (in terms of Schur functions) is given by
\begin{equation}...
- 2,137
8
votes
1
answer
387
views
Interaction of plethysm with other operations
The plethysm $s_{\nu}[s_{\mu}]$ of two symmetric functions is the character of the composition of Schur functors $S^{\nu}(S^{\mu}(V))$. We know that this operation is linear and multiplicative in its ...
- 891
6
votes
1
answer
176
views
On a certain expansion in term of Schur functions
This question is related to this other one
A Schur positivity conjecture related to row and column permutations
by Richard Stanley (thanks to Sam Hopkins for letting me know about it).
Consider a ...
1
vote
0
answers
138
views
Conjugation of bosonic and fermionic
We use the notation from semi-infinite wedge formalism $\bigwedge^{\infty/2}V$ with vector space $V$ generated by $$\left\{\underline{s}\mid s \in \mathbb{Z}+\frac12\right\}$$,
we consider the charge ...
- 685
2
votes
0
answers
104
views
Cut and Join for Hurwitz number with multiple spin
Let me introduce some background of cut and join equation for spin Hurwitz number with the completed cycle as mentioned in
https://arxiv.org/pdf/1103.3120.pdf
We fix two partition $\mu $ and $\nu$ of ...
- 685
7
votes
1
answer
376
views
Jack function in power symmetric basis
In Macdonald's book, the Jack symmetric function $J_{\lambda}(x_1,\ldots, x_n)$ for a partition $\lambda$
is defined by three properties (orthogonality, triangularity, and normalization). In the ...
- 685
11
votes
2
answers
515
views
Using irreducible characters of the orthogonal group as basis for homogeneous symmetric polynomials
The irreducible characters of the orthogonal group $O(2N)$ are given by
$$ o_\lambda(x_1,x_1^{-1},...x_N,x_N^{-1})=\frac{\det(x_j^{N+\lambda_i-i}+x_j^{-(N+\lambda_i-i)})}{\det(x_j^{N-i}+x_j^{-(N-i)})}...
- 1,549
7
votes
2
answers
389
views
What makes skew characters of the symmetric group special?
For integer partitions $\mu\subset\lambda$ we can define the skew character $\chi^{\lambda/\mu}$ (for example?) via the Littlewood-Richardson rule.
Many combinatorial gadgets and algorithms extend in ...
- 5,822
4
votes
1
answer
208
views
Applying a simple involution to Hall-Littlewood polynomials
Consider the Hall-Littlewood polynomial
$$
P_\lambda(x_1,\ldots,x_n;t)=\sum_{\sigma\in S_n/S_n^\lambda}\sigma\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod\limits_{\lambda_i>\lambda_j}\dfrac{x_i-...
- 2,516
3
votes
0
answers
82
views
Macdonald's idea of his kth weight
This question is about Macdonald's symmetric polynomials theory. Going through related papers and literature, it seems to me that the magical part of his theory lies in how the k-th weight function $\...
- 5,230
3
votes
0
answers
203
views
Decomposing Schur functor applied to a tensor product
I want to compute
$$
S^{2,2,\dots,2,1}(\mathbb C^{2m-1} \otimes W)^{SL(2m-1)}
$$
Here $m$ numbers should appear in the superscipt of the Schur functor, and the last superscript means to take $SL(2m-1)$...
- 1,509
2
votes
0
answers
87
views
Schur function on unit circles
Define $T^d$ as following
$$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{C}^{d}\mid |t_i|= 1 \mbox{ for all } i\right\}
$$
For any partition $\lambda\vdash n$,The Schur function is defined
$$
\...
- 1,503
9
votes
2
answers
772
views
Characters of orthogonal groups as symmetric functions
This question was asked on MSE some time ago, here, but got no attention.
The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)=Tr(R_\lambda(U))$. They are symmetric ...
- 2,552
10
votes
3
answers
828
views
The vanishing of sum of coefficients: symmetric polynomials
Denote $\pmb{X}_n=(x_1,x_2,\dots,x_n)$. Consider the symmetric polynomial
$$f_n(\pmb X_n)=\prod_{1\leq i<j\leq n}(x_i+x_j).$$
Expand these in terms of elementary symmetric polynomials, say
$$f_n(\...
- 43.2k
4
votes
1
answer
202
views
Littlewood-Richardson coefficients for zonal polynomials
The Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ appear in the expansion of a product of Schur functions into Schur functions, $s_{\mu}(x)s_\nu(x)=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda(x)...
- 2,552
15
votes
1
answer
748
views
Schur-Weyl duality and q-symmetric functions
Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question ...
- 7,789
13
votes
1
answer
399
views
Is there a Giambelli identity with dual representations?
For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition.
...
- 148k
2
votes
0
answers
99
views
Diagonal operator and infinite wedge space formalism
Let $\bigwedge^{\infty /2}V$ denote semiinfinte wedge space. The followin article section 2 gives a good description about the space and the operator on it.
https://arxiv.org/pdf/math/0207233.pdf
...
- 685
15
votes
1
answer
749
views
Character theoretic proof of the Littlewood–Richardson rule?
The Littlewood–Richardson coefficients are the multiplicities
$$
c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu))
$$
and the Littlewood–Richardson rule says that ...
- 1,413