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.
370 questions
41 votes
3 answers
2k views
Characterizing positivity of formal group laws
The formal group law associated with a generating function $f(x) = x + \sum_{n=2}^\infty a_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my thesis, I found a large number of examples of ...
- 538
36 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 ...
25 votes
1 answer
868 views
Only finitely many values of the symmetric functions of $1/1,1/2,\ldots,1/n$ are $2$-adic integers (?)
For integers $n \geq k \geq 1$ let $$H(n,k) := \sum_{1 \leq i_1 < \cdots < i_k \leq n} \frac1{i_1 \cdots i_k}$$ be the $k$-th elementary symmetric function of $\tfrac1{1},\tfrac1{2}, \ldots, \...
user40023
23 votes
5 answers
3k 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 $\lambda$ ...
- 11.7k
21 votes
4 answers
8k views
Expressing power sum symmetric polynomials in terms of lower degree power sums
Is there an explicit formula expressing the power sum symmetric polynomials $$p_k(x_1,\ldots,x_N)=\sum\nolimits_{i=1}^N x_i^k = x_1^k+\cdots+x_N^k$$ of degree $k$ in $N < k$ variables entirely ...
- 213
21 votes
1 answer
2k views
Why are the power symmetric functions sums of hook Schur functions only?
One interesting fact in symmetric function theory is that the power symmetric function $p_n$ can be written as an alternating sum of hook Schur functions $s_{\lambda}$: $$ p_n = \sum_{k+\ell = n} (-1)^...
- 1,800
21 votes
1 answer
1k views
Bounding Schur symmetric polynomials on the unit circle
Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by \begin{equation} s_\lambda(x_1,\ldots, x_n) = a_{\lambda + \delta}(x_1, \...
- 4,496
19 votes
2 answers
2k views
What role does Cauchy's determinant identity play in combinatorics?
When studying representation theory, special functions or various other topics one is very likely to encounter the following identity at some point: $$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \...
- 86.4k
19 votes
1 answer
765 views
Simple proof that certain walks in the plane don't intersect
Suppose that $n$ hamsters are at the points $(1,n)$, $(2,n),\dots, (n,n)$ in the plane. They walk independently one step east with probability $p$ or one step south with probability $1-p$, until ...
- 53.6k
19 votes
1 answer
1k 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
18 votes
0 answers
491 views
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$ ...
- 53.6k
18 votes
0 answers
422 views
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_\...
- 28.2k
17 votes
4 answers
2k views
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 ...
- 4,496
17 votes
1 answer
3k views
Maclaurin's inequality on elementary symmetric polynomials of arbitrary real numbers
Is there a universal constant $C$ such that the following statement holds? For concreteness, you may assume $C=10000$. Let $a = (a_1, \ldots, a_n)$ be $n$ arbitrary real numbers. For an integer $k$, ...
- 381
17 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 ...
- 35.5k
17 votes
1 answer
781 views
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 ...
- 3,841
16 votes
1 answer
915 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 ...
- 2,907
16 votes
1 answer
861 views
A symmetric function related to sums of square roots
Let $x_1,x_2,\dots,x_n$ be indeterminates (say over $\mathbb{Q}$). For every sequence $\epsilon=(\epsilon_1, \dots,\epsilon_n)\in\{-1,1\}^n$ define $$ y_\epsilon = \sum_i \epsilon_i \sqrt{x_i}. $$ Let ...
- 53.6k
16 votes
0 answers
296 views
Generalization of Newton's identities to Schur functions
In some recent work, I've stumbled across the following identity for $\lambda \vdash n$: $$ n s_\lambda = \sum_{k=1}^n p_k \sum_{\mu \nearrow_k \lambda} (-1)^{\mathrm{ht}(\lambda/\mu)} s_\mu. $$ Here, ...
- 2,009
16 votes
1 answer
742 views
Some questions related to meanders
Let $A_n$ denote the set of noncrossing fixed point free involutions in the symmetric group $S_{2n}$. "Noncrossing" means that if $a<b<c<d$, then not both $(a,c)$ and $(b,d)$ can be ...
- 53.6k
15 votes
2 answers
896 views
An orbit of symmetric polynomials
Consider the ring of polynomials $R:=\mathbb{Z}[x_1,x_2,x_3]$. Define the operators $E, I:R\rightarrow R$ by $Ef(x_1,x_2,x_3)=f(x_1-1,x_2,x_3)$ and the identity $If=f$. Let $\mathcal{L}:R\rightarrow R$...
- 43.7k
15 votes
1 answer
820 views
Schur-Weyl duality and q-symmetric functions
Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question ...
- 7,999
15 votes
1 answer
322 views
A formula for this generating function that is similar to the $qt$-Catalan numbers
I came up with the following conjecture: $$ \sum_{n \ge 0} z^n \sum_{\mu \vdash n} \frac{ t^{\sum l}q^{\sum a}}{\prod (q^a - t^{l+1})(t^l - q^{a+1})} = \exp\left(\sum_{n \ge 1} \frac{z^n}{n(q^n-1)(t^n-...
- 1,549
15 votes
1 answer
824 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 ...
- 1,443
14 votes
2 answers
2k views
Status of the Stanley–Stembridge conjecture
As mentioned in the post on Stanley's 25 positivity problems, Tatsuyuki Hikita posted a preprint on October 16, 2024 purporting to prove Problem 21, the Stanley–Stembridge conjecture about e-...
- 1,026
14 votes
3 answers
678 views
Schubert calculus expressed in terms of the cotangent space of the Grassmannians
Let $T^*_{\mathbb{C}}(Gr_{n,r})$ denote the cotangent space of the Grassmannian of $r$-planes in $\mathbb{C}^n$. Moreover, let $\Lambda^\bullet$ denote the exterior algebra of $T^*_{\mathbb{C}}(Gr_{n,...
- 443
14 votes
1 answer
716 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(...
- 86.4k
14 votes
1 answer
704 views
A Schur positivity conjecture related to row and column permutations
The problem Counting cycles after permuting within rows and columns reminds me of the following unpublished conjecture of mine. Let $D$ be any finite planar diagram (in the sense of Young diagram, ...
- 53.6k
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 ...
- 35.5k
14 votes
1 answer
539 views
Inequalities on elementary symmetric polynomials
I have recently come across the following result. Let $0 < d \leq n$. Given any vector $x \in \mathbb{R}^n$ that satisfies $e_{d-1}(x) = 0$, show that $$|x_1 \cdots x_d| \leq |e_d(x)|$$ where $...
- 1,247
14 votes
1 answer
809 views
a Vandermonde-type of determinants summed over permutations
Let $S_n$ be the symmetric group. Consider $$D:=\sum_{\sigma\in S_n} \text{sgn}(\sigma)\cdot \det\begin{pmatrix}1 & a_{\sigma(1)}-0 & (a_{\sigma(1)}-0)^2 & \cdots & (a_{\sigma(1)}-0)^{...
- 141
14 votes
2 answers
1k views
Symmetric group action on squarefree polynomials
The following dynamical system on polynomials comes mostly from idle curiosity, but I hope it is of some interest. Background Fix some natural number $n$. Let $P$ be the quotient of the polynomial ...
- 15.9k
13 votes
3 answers
1k views
Examples of specializations of elementary symmetric polynomials
Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} e_{h}(\...
- 231
13 votes
1 answer
1k views
Can the Jacobi-Trudi identity be understood as a BGG resolution?
The thought process that led me to this question is that the identity $$ \left(\prod_i \frac1{1-x_i}\right)\left(\prod_i {1-x_i}\right)=1$$ can be understood as expressing exactness of the Koszul ...
- 9,192
13 votes
1 answer
444 views
Is there a Giambelli identity with dual representations?
For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition. ...
- 162k
13 votes
4 answers
1k views
$S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial
Let $S_k$ be the $k$-th elementary symmetric polynomial of $n$ variables. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and $y=(...
- 181
13 votes
1 answer
892 views
Most computationally efficient Littlewood-Richardson rule
There are many, many different versions of the Littlewood-Richardson rule: the original characterization via Yamanouchi words, Remmel's version, a description via the Poirier-Reutenauer bialgebra, the ...
- 2,009
13 votes
1 answer
477 views
Counting higher-dimensional partitions with symmetric function theory
My coauthors and I are writing a (mostly expository) paper in which we construct the Specht module. Our proof that the Specht module is irreducible in characteristic zero implies the following ...
- 5,958
13 votes
1 answer
276 views
Recognizing algebraic independence among Schur polynomials
Given a set of integer partitions $\{\lambda_1, \lambda_2,\dots \lambda_n\}$. Are there combinatorial criteria for deciding whether the associated Schur polynomials $s_{\lambda_1}, s_{\lambda_2},\dots ...
- 86.4k
12 votes
3 answers
531 views
Integer-valued power sums
Suppose I have a positive number $d \in \mathbb{R}$ and a sequence of numbers $a_n \in [0,d]$ for $n \in \mathbb{N}$ with the following properties $$ \sum_{i=1}^{\infty} a_i^r \in \mathbb{Z} $$ for ...
- 7,677
12 votes
3 answers
1k views
Symmetric version of Hilbert's seventeenth problem?
Artin's solution to Hilbert's seventeenth problem tells us that a multivariate polynomial $f$ takes only non-negative values over the reals if and only if it is a sum of squares of rational functions. ...
- 88.2k
12 votes
1 answer
929 views
Plugging $1-x$ into Schur polynomials
I have a symmetric Laurent polynomial $f$ in $k$ variables expressed as a linear combination of Schur polynomials. I'd like to know what happens when I make the substitution $p(x_1,\ldots,x_k)\mapsto ...
- 1,530
12 votes
2 answers
755 views
On shifted symmetric power sums
The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define $p^*_{(k_1,k_2,...)}=p^*_{k_1}p^*_{...
- 2,572
12 votes
1 answer
821 views
Is the appearance of Schur functions a coincidence?
The Schur functions are symmetric functions which appear in several different contexts: The characters of the irreducible representations for the symmetric group (under the characteristic isometry). ...
- 193
12 votes
2 answers
561 views
Dynamics of RSK
There is a way of viewing the RSK correspondence as a map (in fact, bijection) $A \overset{RSK}\longrightarrow \widehat{A}$ from $n\times n$ matrices with entries $\mathbb{N}$ to (weak) reverse plane ...
- 25.9k
11 votes
2 answers
625 views
Using irreducible characters of the orthogonal group as basis for homogeneous symmetric polynomials
The irreducible characters of the orthogonal group $O(2N)$ are given by $$ 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
2 answers
361 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 $n^2 \cdot d$ ...
- 1,194
11 votes
4 answers
696 views
Heisenberg algebra from Pieri operators and their transposes?
Let $Symm$ be the vector space with basis $(b_\lambda)$ given by the set of all partitions $\lambda$ (of all natural numbers), thought of as Young diagrams. Let $e_i$ be the degree $i$ Pieri operator ...
- 28.2k
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 $...
- 5,012
11 votes
1 answer
702 views
Plethysm of $\mathrm{QSym}$ into $\mathrm{QSym}$: can it be defined?
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 ...
- 35.5k