All Questions
Tagged with rt.representation-theory symmetric-functions
99 questions
35
votes
5
answers
6k
views
Understanding a quip from Gian-Carlo Rota
In the chapter "A Mathematician's Gossip" of his renowned Indiscrete Thoughts, Rota launches into a diatribe concerning the "replete injustice" of misplaced credit and "forgetful hero-worshiping" of ...
23
votes
5
answers
2k
views
Is there a short proof that the Kostka number $K_{\lambda \mu}$ is non-zero whenever $\lambda$ dominates $\mu$?
This is maybe a little basic for MathOverflow, but I'm hoping it will get some interesting answers.
Let $\unrhd$ be the dominance order on partitions of $n \in \mathbb{N}$.
For partitions $\lambda$ ...
- 11.2k
21
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, \...
- 4,466
19
votes
1
answer
889
views
What is currently known or conjectured about q,t-Kostka polynomials?
The $q,t$-Kostka polynomials $K_{\lambda,\mu}(q,t)$ appear as the change of basis coefficients between Macdonald polynomials $H_\mu(x;q,t)$ and Schur functions $s_\lambda(x)$:
$$H_\mu(x;q,t)=\sum_{\...
- 307
15
votes
1
answer
748
views
Schur-Weyl duality and q-symmetric functions
Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question ...
- 7,789
15
votes
1
answer
749
views
Character theoretic proof of the Littlewood–Richardson rule?
The Littlewood–Richardson coefficients are the multiplicities
$$
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
14
votes
2
answers
2k
views
Diagonal invariants of the symmetric group on $k[X_1,X_2,...,X_n,Y_1,Y_2,...,Y_n]$
This sounds like something that must have been answered long ago, but for some reason I can find nothing on it in the internet. (There has been lots of recent activity in diagonal covariants, related ...
- 33.8k
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
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 ...
- 33.8k
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
12
votes
1
answer
846
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 ...
- 1,510
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
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
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 $...
- 4,952
10
votes
2
answers
2k
views
Jack polynomials as determinants
Jack symmetric polynomials are known to be generalizations of Schur functions $\chi_\lambda$, for which powerful Weyl determinant formulas are known.
Are there any generalizations of two determinant ...
- 1,343
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
$$f_n(\...
- 43.2k
10
votes
1
answer
522
views
Cauchy identity in three sets of variables?
The Cauchy identity states that
$$
\prod_{i,j} \frac{1}{1-x_i y_j} = \sum_\lambda s_\lambda(x) s_\lambda(y),
$$
where $s_\lambda(x)$ is the Schur function.
Is there a known decomposition of the ...
- 273
10
votes
2
answers
1k
views
Can you find linear recurrence relation for dimensions of invariant tensors?
Let $V$ be a finite dimensional highest weight representation of a (semi)-simple Lie algebra. For each $n\ge 0$ take $a_n$ to be the dimension of the space of invariant tensors in $\otimes^n V$.
In ...
- 9,132
10
votes
1
answer
413
views
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
10
votes
1
answer
627
views
Littlewood-Richardson coefficients for Jack symmetric functions
Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(\alpha)$.
We define a scalar product $\langle \cdot,\cdot\rangle_\alpha$ on $\Lambda$ by setting $\langle p_\mu,p_\nu\rangle_\...
- 158
10
votes
1
answer
437
views
Reference request: Grothendieck groups of Hecke algebras at root of unity and symmetric functions
Let $\zeta$ be an $\ell^{\text{th}}$ root of unity, and consider $H_n(\zeta)$, the (finite) Hecke algebra of type A. One can consider a dual pair of Hopf algebras arising from this data, denoted $G(\...
- 485
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
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
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
8
votes
1
answer
1k
views
Details about plethysm
I'm currently working on plethysm, i.e. the character of the composition $S^\lambda(S^\mu(V))$ of the Schur functors $S^\lambda$ and $S^\mu$ on a complex vector space $V$. We note this character $s_\...
- 891
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
8
votes
1
answer
1k
views
A combinatorial expression of Hall-Littlewood polynomials
This is related to the question Hall-Littlewood functions and functions on the nilpotent cone, and arises in the construction of Coulomb branches of gauge theories. The motivation is explained at the ...
- 3,051
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
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
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
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
7
votes
2
answers
778
views
Can one recognize this symmetric function?
$\newcommand{\lm}{\lambda}$ $\newcommand{\bR}{\mathbb{R}}$ Let $m$ be an integer $>1$. Define
$$ I_m:\bR^m\to \bR,\;\; I_m(\lm_1,\dotsc, \lm_m)=\int_{S^{m-1}}\exp\Bigl(-\sum_{j=1}^m \lm_j^2x_j^2\...
- 34.7k
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
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
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
7
votes
1
answer
743
views
schur weyl duality for real orthogonal groups and relation to hyperoctahedral groups
I am wondering whether the Lie groups $SO(n)$ and the hyperoctahedral groups $H_n$ form some sort of duality. I am mainly interested in how to parametrize the conjugacy classes of $H_n$ in terms of ...
- 4,466
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
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
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
7
votes
0
answers
261
views
Explicit form of raising and lowering operators in spherical gl(n) DAHA
I am working with polynomial representations of spherical subalgebra of double affine Hecke algebra (DAHA) for $\mathfrak{gl}_n$.
Let's call this algebra $\mathfrak{A}_n$ for short. Typically we think ...
- 765
7
votes
0
answers
138
views
Skew zonal polynomials, skew zonal spherical functions, and combinatorics
Zonal polynomials may be expressed in terms of power sums as $$Z_\lambda=n!\sum_\nu \frac{1}{z_\nu }2^{n-\ell(\nu)}\omega_\lambda(\nu)p_\nu,$$ with usual notation in which $\omega_\lambda(\nu)$ are ...
- 2,552
6
votes
3
answers
1k
views
Symmetric powers of Schur polynomials
I apologize if this question is trivial, but a couple of days of searching for necessary routines have led me here.
Does there exist software to compute symmetric powers of Schur polynomials?
I ...
- 247
6
votes
1
answer
176
views
On a certain expansion in term of Schur functions
This question is related to this other one
A Schur positivity conjecture related to row and column permutations
by Richard Stanley (thanks to Sam Hopkins for letting me know about it).
Consider a ...
6
votes
2
answers
429
views
Does the ring generated by the odd power sum symmetric functions have a name?
Let $\Lambda$ be the ring of symmetric functions and recall the power sum symmetric function $p_i = \sum x_1^i + x_2^i + \dots$ generate this ring. Let $\tilde\Lambda$ be the ring generated by the odd ...
- 1,989
6
votes
1
answer
186
views
Action of $\widehat{\mathfrak{sl}_2}$ on symmetric functions with $\mathbb{Z}_{(2)}$ coefficients
It is known that there is a representation of the affine Lie algebra $\widehat{\mathfrak{sl}_q}$ (over $\mathbb{Z}$) on the algebra of symmetric functions, where the action of the Chevalley generators ...
- 3,446
6
votes
1
answer
778
views
Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$
I need some help about the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ E(f):=\sum_{j=1}^{...
- 307
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
$$
R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
- 528
6
votes
0
answers
246
views
What do Macdonald polynomials hint about $\operatorname{Rep}(S_\infty)$?
$\DeclareMathOperator\Rep{Rep}$It is well-known that the Macdonald "$P$" polynomials deform the
Jack "$J$" polynomials [1]. The latter have profound relations with
representation ...
- 5,230
6
votes
0
answers
124
views
Natural maps between Schur functors: understanding the image
Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map
$$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$
Let $[\Lambda^2 V]...
- 4,316