Questions tagged [algebraic-combinatorics]

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A combinatoric identity for characters of reductive groups

Let $G$ be a reductive group over an algebraic closed field (of char 0 if necessary). Let $T\subset G$ be a maximal torus and $S=\mathrm{Sym}^*(X(T))$ be the symmetric algebra of characters of $T$. ...
Fyy's user avatar
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12 votes
0 answers
749 views

Does this matrix norm inequality have interesting application in other areas of mathematics?

In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices: Theorem 3. ‎Let $A=[a_{ij}]$ be a real symmetric ...
Mostafa's user avatar
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36 votes
1 answer
1k views

Errata for Fulton's "Young tableaux"

Fulton's Young tableaux is one of the best texts on the subject, one which I often recommend and cite for reference. Unlike Fulton/Lang and Fulton/Harris, it is neither an early-dawn draft nor a ...
3 votes
1 answer
223 views

Sum of squares of chromatic roots of a bipartite graph

Given a graph $G = (V, E)$, we can calculate its chromatic polynomial $P(G, k)$, and it has $n$ (complex) roots, also known as chromatic roots. It is a well-known fact that the sum of chromatic roots ...
hedgehog0's user avatar
1 vote
0 answers
135 views

References/applications/context for certain polytopes

First, let's consider an almost trivial notion. With any subspace $V\subset \mathbb R^n$ we associate a convex polytope $P(V)\subset V^*$ as follows. Each of the $n$ coordinates in $\mathbb R^n$ is a ...
Igor Makhlin's user avatar
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0 votes
0 answers
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Gessel-Viennot theorem

In the paper, page 76, why we need the condition that the subpath lying between lines y=-x and y=k+1 consists entirely of vertical steps?
Mihawk's user avatar
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7 votes
0 answers
91 views

Pattern avoidance and P-recursiveness

A sequence $\{a_n\}_{n \geq 0}$ is said to be P-recursive if there exist polynomials $p_0(n), p_1(n), \dots , p_k(n)$ such that $$ \sum_{i=0}^k p_i(n) a_{n+i}=0 $$ for all $n \in \mathbb N$. Let $ P$ ...
Pluviophile's user avatar
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4 votes
1 answer
306 views

Determining when quotient of a polynomial ring is a Gorenstein ring

I would like to be able to look at the ring $R=\mathbb{Z}[x_1,x_2,\ldots,x_n]/\mathcal{I},$ where $\mathcal{I}$ is generated by a finite number of monomials and say whether $R$ is a Gorenstein ring. ...
Haldot's user avatar
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5 votes
0 answers
137 views

How did Macdonald come up with $q,t$-Kostka polynomials?

The $q,t$ Kostka polynomials are defined to be the coefficients of the big Schur $s_\lambda[X(1-t)]$ in the expansion of the integral form Macdonald polynomials $J_\mu[X;q,t]$. The integral form ...
ArB's user avatar
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7 votes
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Mistakes in Logan and Shepp's famous paper on Young Tableaux?

In their landmark paper from 1977 named "A Variational Problem for Random Young Tableaux" Logan and Shepp obtained a number of results concerning asymptotic properties of Young Diagrams ...
Matteo's user avatar
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4 votes
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190 views

Schur polynomials are polynomials but also sequences on a lattice?

Monomial symmetric polynomials in $n$ variables $x_1, \ldots x_n$ form a natural basis for the space $\mathcal{S}_n$ of symmetric polynomials in $n$ variables and are defined by additive ...
Arnold Mckenzie's user avatar
0 votes
0 answers
283 views

Relation between $3$-term Plücker relations and more than $3$-term Plücker relations

$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n)$ be the Grassmannian variety of $k$-planes in an $n$-dimensional vector space. The coordinate algebra $\mathbb{C}[\Gr(k,n)]$ is generated by Plücker ...
Jianrong Li's user avatar
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2 votes
1 answer
250 views

Identities involving Littlewood–Richardson coefficients?

I am not aware of that many identities that involve several Littlewood–Richardson coefficients. One recent identity, is a generating function as sum of squares of LR-coefficients, due to Harris and ...
Per Alexandersson's user avatar
15 votes
1 answer
566 views

What is the centralizer of a Young subgroup of $S_n$?

In their celebrated paper "A new approach to the representation theory of the symmetric group. II", Okounkov and Vershik prove that $Z(n-1,1)$, the centralizer of $\mathbb{C}[S_{n-1}]$ in $\...
Alvaro Martinez's user avatar
8 votes
0 answers
122 views

Conceptual explanation for the gap in the spectrum of biregular trees

Ramanujan graphs are defined as $(q+1)$-regular graphs for which all nontrivial eigenvalues of the adjacency operator are contained in the interval $$[-2\sqrt{q}, 2\sqrt{q}].$$ The reason for this ...
Antoine Labelle's user avatar
6 votes
1 answer
176 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 ...
Antoine Labelle's user avatar
8 votes
0 answers
309 views

Why are these Littlewood-Richardson coefficients congruent to 1 mod 8?

Let $n\in{\mathbb N}$ and write $n=q_1+q_2+\dots+q_t$, where $q_1>q_2>\dots>q_t$ are powers of $2$. Let $\lambda_n$ be the partition with Frobenius symbol $(q_1-1,q_2-1,\dots,q_t-1;q_t,q_{t-1}...
John Murray's user avatar
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9 votes
0 answers
229 views

Hives for other root systems? [duplicate]

Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ have numerous combinatorial interpretations, including the hive model by Knutson and Tao, see here. For other root systems, there are also ...
Igor Pak's user avatar
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8 votes
0 answers
224 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_\...
Richard Stanley's user avatar
0 votes
0 answers
88 views

Addition theorem for Schur function in multivariable

Working with the following problem Expansion in Schur function of negative binomial exponent I need to find the expansion of $$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$ in terms of schur ...
GGT's user avatar
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3 votes
0 answers
132 views

Expansion in Schur function of negative binomial exponent

I want to know if there exist a known expansion or can be derived of the polynomial $$ \prod_{i=1}^{m}\prod_{j= 1}^{n}(1-z(x_i + y_i))^{-w} \tag{*}$$ in terms of Schur function. That is asking for (*) ...
GGT's user avatar
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5 votes
1 answer
130 views

PBW basis for the quantized enveloping Lie algebra of $\mathfrak{g}_2$

I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig. Alternatively I ...
Alessandro Carotenuto's user avatar
3 votes
0 answers
67 views

Subrings of the ring of symmetric functions

While experimenting with symmetric functions, I noticed the following equality of subrings of the ring of symmetric functions: $$\mathbb{Z}[(n-1)!\cdot p_n \ |\ n \ge 1] = \mathbb{Z}[n!\cdot h_n \ |\ ...
Antoine Labelle's user avatar
2 votes
3 answers
394 views

Question for averaging the overall quantities by averaging a part

There is a question: If integers $a$ and $b$ satisfy the following properties: for any $a$ real numbers, we can do an operation to average $b$ of them to the same quantities, and we can do a finite ...
JetfiRex's user avatar
  • 511
15 votes
2 answers
632 views

Do power sums determine the variables?

In my analysis research, I came across the following problem. Given $n$ positive real numbers $x_1,\dots,x_n$, consider the $n$-many power sums $$ p_3 = x_1^3 + x_2^3 + \dots + x_n^3 , $$ $$ p_5 = ...
Thierry Laurens's user avatar
5 votes
1 answer
786 views

Expressing symmetric function in power-sum basis

I am trying to prove the following identity \begin{equation} \prod_{i=1}^{m}(1-x_{i}z)^{-u}\prod_{j=1}^{n}(1-y_{i}z)^{-v} \prod_{i=1}^{m}\prod_{j=1}^{n}(1-(x_i +y_j)z)^{-w}\\ = \sum_{\lambda, \mu}c_{\...
GGT's user avatar
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0 answers
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Algebraic ode of exponential generating series

Let $G(z)$ be a rational function. So if we have a series $$S(x):=\sum_{n}a_n x^n $$ where $$ a_n = \prod_{i=1}^{n}G((i-1)h) $$ We can conclude that the series satisfies a Linear differential ...
GGT's user avatar
  • 685
4 votes
2 answers
254 views

A special class of weighted Motzkin paths

Consider Motzkin paths with the following weight: All up-steps and the horizontal steps on height $0$ have weight $1$, all down-steps have weight $t$ and the horizontal steps on even heights have ...
Johann Cigler's user avatar
2 votes
0 answers
194 views

Geometric or combinatorial interpretations of the (weak) Bruhat order?

$\DeclareMathOperator\Inv{Inv}$The weak Bruhat order on the symmetric group has a straightforward combinatorial interpretation: Consider a set of labelled balls $1,2,\dotsc,n$. Then for two ...
Brendan Mallery's user avatar
10 votes
1 answer
291 views

$2$-adic valuation of Schur $P$-functions in the power-sum basis

For a partition $\lambda$, let $P_\lambda$ be the Schur $P$-functions (case $t=-1$ of Hall-Littlewood symmetric functions) and let $p_\lambda=p_{\lambda_1}p_{\lambda_1}\cdots p_{\lambda_k}$ be the ...
Antoine Labelle's user avatar
20 votes
2 answers
887 views

The finite groups with a zero entry in each column of its character table (except the first one)

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Aut{Aut}$Consider the class of finite groups $G$ having a zero entry in each column of its character table (except the first one), i.e. for all $g \...
Sebastien Palcoux's user avatar
0 votes
1 answer
257 views

What is a toric lattice? [closed]

What is a toric lattice? and how can I construct one in Macaulay2 and compute its basis? is there any alternative method to make one? Since I went through the whole ...
iMan's user avatar
  • 13
7 votes
0 answers
160 views

Littelmann Path model and RSK e and f operators

The Littelmann path model defines $e_i$ and $f_i$ operators which correspond in type A to the $e_i$ and $f_i$ operators on semistandard Young tableaux (i.e. since they both are different ways of ...
Mathprof's user avatar
  • 171
8 votes
1 answer
251 views

Branching rule for Specht modules over Kazhdan-Lusztig basis

Suppose $\lambda\vdash n$ is a partition and $S^\lambda$ is the associated irreducible representation of $S_n$. As we know from the branching rule, we have an isomorphism of $S_{n-1}$ modules $$S^\...
GossipM's user avatar
  • 221
6 votes
1 answer
501 views

Can Matsumoto's theorem for the symmetric group be proved using a monovariant?

This is a question that can be asked for any Coxeter group, but for the sake of simplicity I will restrict myself to symmetric groups. Recall the main definitions: Let $n$ be a nonnegative integer. ...
darij grinberg's user avatar
0 votes
0 answers
136 views

How to classify rings by combinatorial structures?

There are many ways to encode information about algebraic structures such as groups, rings, etc... in combinatorial form. For example the Cayley graph of a group with a subset of generators, or the ...
Vni Versvs's user avatar
7 votes
2 answers
450 views

Chip-firing clocks

Let $G$ be some outdegree-regular directed graph with $n$ vertices and let $H$ be the Laplacian of $G$, so that the rows of $H$ correspond to chip-firing moves. I’m interested in linear functions $f$ ...
James Propp's user avatar
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1 vote
0 answers
84 views

Making the entries of a matrix positive

I am considering two slightly more relaxed version of the question asked here: https://math.stackexchange.com/questions/119034/making-the-entries-of-a-matrix-positive The two questions are: Question 1:...
Jandré Snyman's user avatar
15 votes
1 answer
708 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)^{...
Fan Ge's user avatar
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9 votes
0 answers
480 views

Two majs for standard Young tableaux?

Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...
Sam Hopkins's user avatar
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4 votes
1 answer
237 views

proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions"

I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...
dash1729's user avatar
3 votes
0 answers
203 views

Does every finite lattice embed into a finite Eulerian lattice?

A finite Boolean lattice is a lattice isomorphic to the subset lattice of a finite set. Every Boolean lattice is Eulerian, namely, a graded lattice $L$ such that $\mu(a,b) = (-1)^{|b|-|a|}$ for all $a,...
Sebastien Palcoux's user avatar
5 votes
0 answers
97 views

Form on symmetric functions and their q,t- analogues

[Notations are as in Macdonald's Symmetric Functions and Hall Polynomials] The space of symmetric functions $\Lambda_{\mathbb{Q}}$ has a bilinear form defined by $ (p_\lambda, p_\mu)= z_\lambda \...
ArB's user avatar
  • 688
2 votes
0 answers
89 views

Double Schur function expansion

In literature, I have seen the weighted Hurwitz number $N_{g,n}(d_1 , d_2 \ldots , d_n)$ which are symmetric number and they can be written as double Schur function expansion. \begin{align} \label{eq:...
GGT's user avatar
  • 685
9 votes
2 answers
501 views

Characterization of the family of simple groups PSL(2,q) by tensor multiplicity

Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$ Let the ...
Sebastien Palcoux's user avatar
3 votes
1 answer
235 views

Generalised operad structures

We can naively consider an operad as a collection $\{P(n)\}_{n\geq 0}$ of vector spaces $P(n)$ consisting of "functions" with $n$ inputs and one output, equipped with a number of ...
Aidan's user avatar
  • 480
6 votes
0 answers
185 views

Word/Loop in $L(\Lambda)$

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody algebra, with Chevalley generators $e_i,f_i$ ($i=1,...,n$). Let $L(\Lambda)$ denote the irreducible module with highest weight $\Lambda$. Let $v$ denote ...
ArB's user avatar
  • 688
3 votes
3 answers
415 views

Given a positive integer $n$, some straight lines and lattice points such... Prove that the number of the lines is at least $n(n+3)$

I was trying to get an answer on MathSE long ago and now I got it. Given a positive integer $n$ and some straight lines in the plane such that none of the lines passes through $(0,0)$, and such that ...
nonuser's user avatar
  • 237
6 votes
1 answer
274 views

Is this Laurent phenomenon explained by invariance/periodicity?

In Chapter 4 (page 23, subsection "Somos sequence update") of his Tracking the Automatic Ant, David Gale discusses three families of recursively defined sequences of numbers, all due to Dana ...
darij grinberg's user avatar
3 votes
0 answers
112 views

Polynomiality of the inverse of equally weighted Varchenko matrices attached to hyperplane arrangements

Let $d\in \mathbb{Z}_{\ge 1}$, let $\sigma = (H_i)_{i\in \mathcal{I}}$ be a finite hyperplane arrangement in $\mathbb{R}^d$, where $H_i\subset \mathbb{R}^d$ is a hyperplane for $i\in \mathcal{I}$ (the ...
Wille Liu's user avatar
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