Questions tagged [symmetric-functions]
Symmetric functions are symmetric polynomials, in finitely many, or countably infinitely many variables. They arise in the representation theory of symmetric groups and in the polynomial representation theory of general linear groups. Bases of the ring of symmetric functions are indexed by integer partitions. Schur functions, elementary symmetric functions, complete symmetric functions, and power sum symmetric functions are the most commonly used bases.
325
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Normalization of Jack polynomial integral-scalar product?
In eq. (10.35) of his book "Symmetric functions and Hall polynomials" I.G.Macdonald gives the following scalar product, under which Jack polynomials with different partitions $\mu\neq\lambda$ are ...
2
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0
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Function expansion in high partition Jack polynomials?
For Jack polynomials $J_\lambda^\alpha(x_1,x_2,...,x_n)$ there exists i.e. the following relation
$$\sum_\lambda J_\lambda^\alpha(x_1,x_2,...,x_n)J_{\lambda'}^{\frac{1}{\alpha}}(y_1,y_2,...,y_m)=\...
9
votes
1
answer
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Hyper-symmetric polynomials (reference request)
Let $M_n$ be the linear space of $n\times n$ matrices. The product of symmetric groups $S_n\times S_n$ acts naturally on $M_n$, and thus induces an action on the coordinate algebra $k[M_n]$. Is there ...
5
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1
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Symmetric power series over $\mathbb{F}_2$
Consider the symmetric power series
$$f = \prod_{i \in I}\left(1+x_i+x_i^2+x_i^4+x_i^8 + x_i^{16} +\ldots \right)$$
in variables $(x_i)_{i \in I}$ over $\mathbb F_2$. Fix some degree $r$, smaller ...
11
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0
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Symmetry of function defined by integral
(Originally posed in Math.SE in Jan 2013. Received no complete answers as of yet.)
Define a function $f(\alpha, \beta)$, $\alpha \in (-1,1)$, $\beta \in (-1,1)$ as
$$ f(\alpha, \beta) = \int_0^{\...
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0
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51
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Need explicit formula for reversion of a Chern-character-like series
On the first sight this looks like homotopy theory question but actually came from need to simplify some expressions related to the Rasch model from the Item Response Theory.
Let
$$
s=\sum_i\frac{...
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2
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157
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Symmetric tensors as sum of powers
I am looking for formulas for writing a basis element of $ Sym^k(H) $ as sum of elements of the form $ v^{\otimes k} $ where $ v\in H $. Here $ H $ is a hilbert space and by basis element I mean the ...
10
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Super-plethysm?
Let $U$ be a representation of $S_m$ and $V$ a representation of $S_n$. Then the representation $\operatorname{Ind}_{S_m\wr S_n}^{S_{mn}}(U^{\otimes{n}}\otimes V)$ has a nice interpretation in terms ...
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Skew zonal polynomials, skew zonal spherical functions, and combinatorics
Zonal polynomials may be expressed in terms of power sums as $$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 ...
5
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2
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sum of squares of Schur polynomials indexed over partition valued functions on a set
Fix a finite set $X$ and two natural numbers $d$ and $n$.
For a partition $\lambda$ and a number $d$ denote by $s_\lambda^d(x_1,\dots,x_d)$ the Schur polynomial in $d$-many variables $x_1,\dots,x_d$. ...
1
vote
1
answer
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Ergodicity of elementary symmetric polynomials with noncommutable variables
Let $\{X_n\}$ be an ergodic sequence of random variables, $X_n:(\Omega,\mathcal{F})\to (S,\mathcal{S})$ where the target set $S$ is a matrix ring. My question is,
Can the following limit be found ...
12
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1
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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 ...
7
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0
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Constant term identity and the Grassmannian Gr(2,6)
The following conjecture is motivated by two different presentations of the affine cone over Grassmannian $Gr(2,6).$ One as a GIT quotient of $Hom(\mathbb{C}^2, \mathbb{C}^6)//SU(2)$ and the other as ...
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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$ ...
6
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0
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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 \...
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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, \...
3
votes
0
answers
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Do any specializations of variables give valid equalities of series and products involving Witt symmetric functions?
Formally, Witt symmetric functions $w_n(x_1,x_2,...)$ ($n\geqslant1$) can be defined by
$$
\prod_n(1-w_nt^n)=1+\sum_k(-1)^ke_kt^k=\prod_j(1-x_jt),
$$
where $e_k(x_1,x_2,...)$ are the elementary ...
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Cohomology of configuration space as a representation of the symmetric group
Let $X_n$ be the space of $n$ distinct labeled points in $\mathbb{R}^3$, which is equipped with an action of the symmetric group $S_n$. It is well known that the total cohomology of $X_n$ is ...
4
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Is there a nice way to invert this expression?
Let us first define the Euler polynomials to be the polynomials $P_n(q)$ that satisfy
$$
\frac{qP_n(q)}{(1 - q)^{n+1}} = \Big(q\frac{d}{dq}\Big)^n\frac{q}{1 - q}.
$$
For example, $P_0(q) = P_1(q) = 1$...
16
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661
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An introduction to Macdonald polynomials other (better?!) than SFHP
Long story short, I personally find Macdonald's celebrated book Symmetric Functions and Hall Polynomials somewhat difficult to read for various reasons. I also know for a fact that I'm not the only ...
13
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1
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674
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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 ...
12
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2
answers
623
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On shifted symmetric power sums
The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define $p^*_{(k_1,k_2,...)}=p^*_{k_1}p^*_{...
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0
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Irreducible representations of $S_n$ inside the ring of symmetric polynomials
I will describe two ways to associate irreducible representations of $S_n$ with polynomials inside the ring of symmetric polynomials and I want to know if there is any connection between the two.
$\...
6
votes
2
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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 ...
18
votes
0
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An algebraic strengthening of the Saturation Conjecture
The Saturation Conjecture (proved by Knutson-Tao) asserts that
$c_{n\mu,n\nu}^{n\lambda}\neq 0\Rightarrow c_{\mu,\nu}^{\lambda} \neq
0$, where $c$ denotes a Littlewood-Richardson coefficient and $n$ ...
2
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183
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integral schur function over standard simplex
Let $T^d$ be the standard simplex,
$$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{R}^{d}\mid\sum_{i = 1}^{d}{t_i} = 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\right\}
$$
For any partition $\lambda\...
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0
answers
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What is the definition of plethysm in the representation theory of permutation groups
Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams.
I have seen the definition of plethysms in symmetric functions. I would like to understand the ...
3
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0
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A "nice" Orthogonal Basis for Translation Invariant Symmetric Polynomials
It is going to be a rather long question, so I will first state it and then try to explain and motivate it.
Take $\Lambda_n $ as the graded ring of symmetric polynomials of a field $F$ in $n$ ...
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0
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164
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Character sums over a fixed subset of skew tableaux
Let $f(\lambda)$ count the number standard young tableaux of shape $\lambda\vdash n$ and $\lambda=(\lambda_1,\cdots,\lambda_r)$. Let $\mu \vdash k$ be a partition for $k<n$. It is a consequence of ...
5
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Uniform generation of Symmetric Plane Partitons
In the conclusion of An Involution Principle-Free Proof of Stanley's Hook-Content Formula Krattenthaler notes that the techniques of the paper might be useful for finding bijective proofs of the ...
6
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1
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Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$
I need some help about the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ E(f):=\sum_{j=1}^{...
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0
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How should this multinomial identity be written?
Question if it is correct, how is the identity tagged (4) below usually written, and can the use of conjugate partitions be avoided?
Motivation I apologize for the length of this question - it's as ...
4
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0
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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 ...
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4
answers
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$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial
Let $S_k$ be the $k$-th elementary symmetric polynomial of $n$ variables. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and $y=(...
1
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1
answer
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Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory
Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let $(n_0,\...
18
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0
answers
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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_\...
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Efficiently computing (plethysm-like?)substitutions of symmetric functions
This is a rather technical question, it arose in connection of some calculations that I need to have better grasp of the question Formal group law over $\mathbb{F}_p$ and my own older one What is ...
2
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counting how many boxes from a given Young tableau contribute to hook length made out of two YTs
Think of a Young diagram as a collection of rows with numbers of elements
$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...
12
votes
2
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Dynamics of RSK
There is a way of viewing the RSK correspondence as a map (in fact, bijection) $A \overset{RSK}\longrightarrow \widehat{A}$ from $n\times n$ matrices with entries $\mathbb{N}$ to (weak) reverse plane ...
3
votes
2
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781
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On a positivity property of Hall-Littlewood polynomials
Here's the new, more thought through version.
Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is ...
4
votes
1
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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 ...
2
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0
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computation with Hilbert scheme of $n$ points on $\mathbb C^2$ [closed]
How can we show that
$$\sum_{n = 0}^\infty q^n \operatorname{char}_T S^n(\mathbb C[x,y])=
\prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$
where $\operatorname{char}_T V$ denotes the character ...
4
votes
1
answer
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Shift-invariant symmetric functions in representation theory?
The connection between symmetric functions and representation theory is well-known.
Now consider the subspace of symmetric functions that are shift-invariant,
that is, functions satisfying $f(x+t+y+t,...
5
votes
2
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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
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1
answer
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Reference or a short argument that a certain subset generates the ring of p-typical symmetric functions under plethysm
Let p be a prime, and let $\Lambda_p$ be the subring of the ring of symmetric functions $\Lambda$ (over $\mathbb{Z}$) such that $$x \in \Lambda_p$$ iff there is an $i \in \mathbb{N}$ such that $p^ix \...
5
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0
answers
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Anti-arithmetic product of symmetric functions: (why) is it integral?
This is an analogue of MathOverflow question #138148. Indeed it is so analogous that I wrote the following by copypasting said question and making the necessary changes.
For every commutative ring $A$...
11
votes
2
answers
651
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A particular specialization of symmetric polynomials: is it bijective?
Let $\Lambda^d_n$ the space of symmetric polynomials
in $n$ variables, with maximum 'partial degree' of each variable $d$.
A basis for this space is the set of symmetrized monomials $m_\lambda$,
where ...
11
votes
3
answers
1k
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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
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0
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82
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The relation on the set of functions
Let $\varphi: \mathbb{R}^{2} \to \mathbb{R}$ be a symmetric (not necessarily continuous) function (so, $\varphi(x,y)=\varphi(y,x)$ $\forall (x,y)\in \mathbb{R}^{2}$),
let $\mathcal{F}$ be the set of ...
8
votes
2
answers
539
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How do I find coefficients of a product expansion
Any power series $f(t) = 1 + t \mathbb{Z}[[t]]$ can be uniquely expanded in the following two ways:
$$1 + \sum_{i=1}^\infty f_i t^i =
\prod_{i=1}^\infty (1-t^i)^{-n_i}$$
Here, the $f_i$ and $n_i$ ...