All Questions
Tagged with symmetric-groups symmetric-functions
40 questions
1
vote
1
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95
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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
9
votes
2
answers
245
views
Matrix invariants for simultaneous conjugation by a finite subgroup of $\textrm{GL}_n$
Let $K$ be a field of characteristic 0, and consider $d$ generic $n\times n$ matrices $X_1,\ldots,X_d$ where $X_k = (x_{ijk})_{ij}$ and $ K[x_{ijk}]$ is the polynomial algebra in $n^2 \cdot d$ ...
- 1,104
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
1
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0
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151
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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
3
votes
0
answers
129
views
Plethysm and wreath product
I am looking for a proof about the link between plethysm and wreath product. It is a well-known fact, being use extensively in many papers, but I can't find a good reference. Everything that follows ...
- 891
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
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0
answers
176
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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
7
votes
1
answer
354
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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
8
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0
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236
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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
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1
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216
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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
8
votes
1
answer
371
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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
1
vote
0
answers
76
views
Determine whether a set generates a residue field of an invariant ring
Fix two positive integers $m>n$.
Let $(A|Y)$ be an $m\times (n+1)$ augmented matrix consisted of $m\times (n+1)$ indeterminates, where $Y$ is a column symbolic vector of length $m$.
Denote $R=\...
- 331
10
votes
0
answers
389
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Has anyone met this "$q$-character" table for $S_4$?
Is anyone aware of the following $q$-character table for the
symmetric group $S_4$?
\begin{array}{|c|c|c|c|c|c|}
\hline
\mathrm{conj}\backslash\mathrm{rep}
& 2+1+1 & 3+1 & ...
- 2,137
6
votes
0
answers
246
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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
1
answer
212
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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
6
votes
1
answer
176
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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 ...
7
votes
2
answers
389
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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
2
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0
answers
90
views
How to prove this identity on summations and partitions?
Let $f$ be a symmetric function of $s$ variables. The identity is
$$\sum_{all \ k's}^\infty f(k_1,k_2,k_3,...,k_s)=\sum_{n=s}^\infty \sum_{\lambda\vdash n}\frac{s!\prod_l \lambda_l}{z_\lambda} f(\...
- 151
2
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0
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135
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A symmetric function that appears in the coefficients of a power expansion
Let's say we have the expression
$$\sum_{k_1=1}^\infty\sum_{k_2=1}^\infty\sum_{k_3=1}^\infty\sum_{k_4=1}^\infty\sum_{k_5=1}^\infty x^{k_1+k_2+k_3+k_4+k_5} f(k_1,k_2+k_3,k_4+k_5)$$
where $f(a,b,c)$ is ...
- 151
15
votes
1
answer
748
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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
15
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1
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749
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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
4
votes
1
answer
814
views
Symmetric functions of eigenvalues
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function, which is symmetric under the action of the symmetric group (acting on $\mathbb{R}^n$ by permuting the variables).
Let $M_{n\times n}$...
- 9,853
6
votes
0
answers
124
views
Natural maps between Schur functors: understanding the image
Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map
$$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$
Let $[\Lambda^2 V]...
- 4,316
1
vote
2
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168
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Rank of a symmetric ideal
Let $\Sigma_m$ be the permutation group on $m$ letters and $R=\mathbb{C}[x_1,x_2,\dots,x_m]$. Let $\Sigma_m$ act on $R$ in the usual way. Let $R^{\Sigma_m}$ denote the ring of $\Sigma_m$ invariant ...
- 422
3
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0
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226
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$S_n$ action on the sequences of transpositions
It is well-known, that any element $\rho$ of the symmetric group $S_n$ with $n-p$ cycles admits a unique presentation as a product of a sequence of transpositions $\{(a_i\,b_i)\}_{i = 1}^p$ with $a_i &...
- 401
10
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1
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413
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Super-plethysm?
Let $U$ be a representation of $S_m$ and $V$ a representation of $S_n$. Then the representation $\operatorname{Ind}_{S_m\wr S_n}^{S_{mn}}(U^{\otimes{n}}\otimes V)$ has a nice interpretation in terms ...
- 2,537
6
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0
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191
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A particular proof of the Littlewood Richardson rule
Given $\lambda \subseteq \nu$ we define a tableau of shape $\nu\setminus \lambda$ and weight $\mu$ to be a map ${\sf T}: [\nu\setminus\lambda] \rightarrow \{1,\ldots, r\}$ such that $\mu_c=|\{ x \...
- 1,413
15
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1
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708
views
Cohomology of configuration space as a representation of the symmetric group
Let $X_n$ be the space of $n$ distinct labeled points in $\mathbb{R}^3$, which is equipped with an action of the symmetric group $S_n$. It is well known that the total cohomology of $X_n$ is ...
- 2,537
2
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0
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674
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What is the definition of plethysm in the representation theory of permutation groups
Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams.
I have seen the definition of plethysms in symmetric functions. I would like to understand the ...
- 349
1
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0
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165
views
Character sums over a fixed subset of skew tableaux
Let $f(\lambda)$ count the number standard young tableaux of shape $\lambda\vdash n$ and $\lambda=(\lambda_1,\cdots,\lambda_r)$. Let $\mu \vdash k$ be a partition for $k<n$. It is a consequence of ...
- 4,952
4
votes
0
answers
1k
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Permutation-invariant matrix representation
The question guide says that Mathoverflow is for research level mathematics. While I do not perform research in mathematics (I study quantum chemistry), I believe this question is research-level ...
- 121
4
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1
answer
149
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Counting a Modified Class of Standard Young Tableau
Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be a partition, with $|\lambda|:=N$. Attach an extra box to $\lambda$ to the right end of the $r$'th row. In coordinate form, the last box on row $r$ has ...
- 4,952
5
votes
2
answers
1k
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Isotypic components of the action of the symmetric group on polynomials
The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S_n$. The isotypic component of the trivial representation is ...
- 2,537
2
votes
1
answer
524
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 ...
- 121
17
votes
4
answers
1k
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universality of Macdonald polynomials
I have been recently learning a lot about Macdonald polynomials, which have been shown to have probabilistic interpretations, more precisely the eigenfunctions of certain Markov chains on the ...
- 4,466
9
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1
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605
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Restriction of characters of hyperoctahedral groups.
The hyperoctahedral group $H_n$ has several descriptions; as a wreath product; as signed permutation matrices; as the Weyl group of type $B_n$ or $C_n$. In all these descriptions it is apparent that ...
- 9,132
3
votes
0
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242
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Picking $n$ so that certain Schur functors of the standard representation of $S_n$ are linearly independent
Let $V_n$ be the standard permutation representation of the symmetric group $S_n$, and let $\mathbb{S}_{\lambda}$ denote the Schur functor associated to the partition $\lambda$.
Let $\lambda$ range ...
- 5,898
2
votes
2
answers
682
views
Convolution on symmetric group Sn
I have question regarding convolution of functions (say
g and h) defined on Sn. In Fourier space this is equivalent to IFT(G.H),
where G = FT(g) and H = FT(h).
Fast Fourier transforms (Clausen's FFT) ...
1
vote
1
answer
642
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Hankel determinants of symmetric functions
The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions.
...
- 9,132