It is well known that the symmetric groups have a very nice and explicit representation theory. This is in particular true when one works with the collection of all symmetric groups simultaneously, in which case the ring of virtual representations is the ring of symmetric functions. This has many advantages: it allows one to reduce hard questions in representation theory to formal combinatorial manipulations with symmetric functions; there is a lot of extra structure like inner product, outer product, and plethysm; it leads directly to the gene...Read more

Let $n$ and $p$ be two positive integers. Consider the function$$\max_{n,p}:\{0,\dots,n\}^p\to\{0,\dots,n\}$$that computes the maximum of a $p$-tuple of integers in the range $\{0,\dots,n\}$. Are there explicit expressions for symmetric polynomials $P_{n,p}\in\mathbb{Q}[\sigma_1,\dots,\sigma_p]$ such that $P_{n,p}(\sigma_1,\dots,\sigma_p)$ interpolates $\displaystyle\max_{n,p}$? Here the $\sigma_i$ are the elementary symmetric polynomials.The case $p=2$ can be done by hand :$P_{n,2}(\sigma_1,\sigma_2)$ can be described by the formula$$\sum_{s=0...Read more

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 symmetric group. To make this post more educational, I will define these polynomials a bit. Consider the 2-parameter family of Macdonald operators (indexed by powers of the indeterminate $X$) for root system $A_n$, on a symmetric polynomial $f$ with $x = (x_1, \ldots, x_n)$:$$D(X;t,q) = a_\delta(x)^{-1} \sum_{\sigma \in S_n} \epsilon(\sigma) x^{\sigma \del...Read more

Background:For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means symmetric power series of bounded degree.) It is known that $\mathbf{Symm}_R$ is generated by the elementary symmetric polynomials $e_1$, $e_2$, $e_3$, ... as an $R$-algebra. If $R$ is a $\mathbb Q$-algebra, then $\mathbf{Symm}_R$ is also generated by the power sum polynomials $p_1$, $p_2$, $p_3$, ... as an $R$-algebra. Note that $\mathbf{Symm}_{\math...Read more

Let $Symm$ be the vector space with basis $(b_\lambda)$ given by theset of all partitions $\lambda$ (of all natural numbers), thoughtof as Young diagrams. Let $e_i$ bethe degree $i$ Pieri operator that takes $b_\lambda \mapsto \sum b_{\lambda'}$ where $\lambda'$ has $i$ moreboxes than $\lambda$, no two in the same column. The $(e_i)$ commute,because they correspond to multiplication by Schur functions inthe ring of symmetric functions.With respect to this basis, we can talk about $e_i^T$,which takes $b_{\lambda'} \mapsto \sum b_{\lambda}$ where...Read more

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.This is motivated by some representation theory.The naive idea is to start with the sequence of symmetric functions $s_{n,n}$ and take the Hankel determinants using the inner product (that is product in the group ring of $S(2n)$) instead of the usual outer product. However this doesn't make sense.Take the $2 \times 2$ case. Then the naive determinant i...Read more

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 $\lambda$. Let $a_i'= a_i +\frac12$ and $b_i'= b_i +\frac12$ are modified Frobenius coordinates.By a classical theorem of Frobenius, it states that $f(\lambda)=\frac{|C_{2,1,\ldots,1}|}{dim \lambda}\chi_{2,1,\ldots,1}^{\lambda}=\frac12\sum_{i=1}^{d(\lambda)}(a_i')^2-(b_i')^2$Notice that the above example we treat the L.H.S as a function of $\lambda$ as ...Read more

I'm wondering if there is a characterization for whether$$p_\lambda(x_1, \dotsc, x_r) \geq p_\mu (x_1, \dotsc, x_r) \text{ for every $r \in \mathbb{N}$ and $x_1, \dotsc, x_r \in \mathbb{R}^+$}.$$(Or where, instead, $x_1, \dotsc, x_r \in \mathbb{N}$.)Here, $p_\lambda$, $p_\mu$ are power sum symmetric functions. That is, $\lambda = \lambda_1, \dotsc, \lambda_n$, $\mu = \mu_1, \dotsc, \mu_m$, $p_\lambda(x_1, \dotsc, x_r) = \prod_i\sum_j x_j^{\lambda_i}$, and $p_\mu(x_1, \dotsc, x_r) = \prod_i\sum_j x_j^{\mu_i}$.We know that when $|\mu|=|\lambda|$...Read more

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 $p_1,p_2,\dots$, where $p_n$ denotes the $n$-th power sum function $x_1^n+x_2^n+\cdots$. Let $\Delta^+$ and $\Delta^{\times}$ denote the $\mathbf{Q}$-algebra maps $\Lambda\to\Lambda\otimes_{\mathbf{Q}}\Lambda$ determined by $\Delta^+(p_n)=1\otimes p_n + p_n\otimes 1$ and $\Delta^{\times}(p_n)=p_n\otimes p_n$ for all $n\geq 1$. Let $\mathbf{Q}_+$ denote the sub-s...Read more

Let $p_1, p_2, \ldots$, be the power sum symmetric functions. Let $p_n^* = n \frac{\partial}{\partial p_n}$. Then $$ p_n^* p_m - p_m p_n^* = \delta_{m, n} 1. $$Where could I find this result in some books or papers? Thank you very much....Read more

Let $\Lambda^d_n$ the space of symmetric polynomialsin $n$ variables, with maximum 'partial degree' of each variable $d$.A basis for this space is the set of symmetrized monomials $m_\lambda$,where $\lambda$ is a partition with maximally $n$ parts, with each part $\leq d$.Take $n= m N$ and $d = N$ (with $m>1$, and both $m$ and $N$ finite)and define the following specialization (or plethysm)$\mathcal{C}$,$$\mathcal{C}: \Lambda^{N}_{m N} \rightarrow \Lambda^{mN}_{N}$$that conflates the $m N$ variables to $N$ variables, via$x_{m(i-1)+j} \righta...Read more

Some will recognize this as similar to a question I asked before, butI want to ask it without the trigonometry.Let $e_k$ be the $k$th-degree elementary symmetric polynomial in$x_1,x_2,x_3,\ldots$. If $k$ is more than the number of $x$s, then$e_k$ is the sum of no terms and is $0$. From one POV, the followingPythagorean identities are as elementary as anything not in thehigh-school curriculum:$$(e_0+e_2+e_4+\cdots)^2 - (e_1+e_3+e_5+\cdots)^2 =(1-x_1^2)(1-x_2^2)(1-x_3^2)\cdots$$$$(e_0-e_2+e_4-\cdots)^2 + (e_1-e_3+e_5-\cdots)^2 =(1+x_1^2)(1+x_2^...Read more

What is known about S_n group invariants in a free associative (noncommutative) algebra k < x_1, ...x_n >) ? (S_n natural acts by permutations on generators).What is Poincare series ? Is it finitely generated, is it free ? Are the generators as algebra/vector space known ? The same question for the commutative algebra - gives algebra of symmetric polynoms,which is for-ever-young research topic. To what extent non-commutative version is the same rich ? PSIs it commutative ? Probably no - however, pay attention on the following simple fact:con...Read more

There are simple expressions for the sums of linear and quadratic combinations of Schur functions over all partitions (including the empty one)$$\sum_\lambda s_\lambda(x)=\prod_{i}\frac{1}{1-x_i}\prod_{i<j}\frac{1}{1-x_ix_j}$$and$$\sum_\lambda s_\lambda(x) s_\lambda(y)=\prod_{i,j}\frac{1}{1-x_iy_j}.$$Let us consider the sum of the ratio of two Schur functions, namely$$\sum_\lambda \frac{s_\lambda(x)}{ s_\lambda(y)}$$Is there any similar expression for this sum?For example, if $\#x_i=\#y_i=1$, we have$$\sum_\lambda \frac{s_\lambda(x)}{ s_\lam...Read more

I will denote by $\Lambda$ the ring of symmetric functions, and by $\mathrm{QSym}$ the ring of quasisymmetric functions (both in infinitely many variables $x_1$, $x_2$, $x_3$, ..., both over $\mathbb Z$). See Victor Reiner, Hopf algebras in combinatorics, chapters 2 and 5, respectively, for the definitions.It is known that one can define a plethysm $f\circ g \in \mathrm{QSym}$ for any $f\in\Lambda$ and any $g\in\mathrm{QSym}$. This is described, e. g., in Claudia Malvenuto, Christophe Reutenauer, Plethysm and conjugation of quasi-symmetric func...Read more