Questions tagged [algebraic-combinatorics]

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25
votes
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815 views

A question on simultaneous conjugation of permutations

Given $a,b\in S_n$ such that their commutator has at least $n-4$ fixed points, is there an element $z\in S_n$ such that $a^z=a^{-1}$, and $b^z=b^{-1}$? Here $a^z:=z^{-1}az$. Magma says that the ...
24
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0answers
836 views

Nekrasov-Okounkov hook length formula

I am now reading the paper An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths by Guo-Niu Han. The author rediscovered what he calls the Nekrasov-...
20
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0answers
348 views

A spin extension of a Coxeter group?

Consider a Coxeter group $W$ with simple generators $S$ and Coxeter matrix $\left( m_{s,t}\right) _{\left( s,t\right) \in S\times S}$. Let $\mathfrak{M}$ be the set of all pairs $\left(s, t\right) ...
18
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0answers
336 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_\...
16
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0answers
440 views

A Product Related to Unrestricted Partitions

Start with the product for unrestricted partitions: $(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$ Now replace some of the plus signs with minus signs and expand the product into a series. Is it ...
13
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0answers
165 views

Relationship between crystal root operators and usual $e_i, f_i$?

Suppose I am working in a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$. Let $e_1,\dotsc,e_n,f_1,\dotsc,f_n$ denote the usual Chevalley generators of $\mathfrak{g}$. Let $V$ be a highest weight ...
12
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0answers
185 views

Non-Boolean Eulerian interval of finite groups

An Eulerian subgroup lattice is Boolean (see here), so it is natural to wonder whether it is also true for an interval of finite groups. The smallest non-Boolean Eulerian lattice is the following: It ...
11
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0answers
555 views

Product of a Schubert polynomial and a double Schubert polynomial

Let $S_u(x)$ be a Schubert polynomial and let $S_v(x;y)$ be a double Schubert polynomial. Then their product can be expressed in terms of the double Schubert polynomials as $$S_u(x)S_v(x;y)=\sum_w{c_{...
10
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0answers
138 views

Branching from $GL(a+b)$ to $GL(a)\times GL(b)$ using Gel'fand-Cetlin patterns

If one iterates the multiplicity-free branching rule from $GL(n)$ representations (finite-dim, over $\mathbb C$) to $GL(n-1)$ all the way down to $GL(0)$, one obtains triangular "Gel'fand-Cetlin (or ...
10
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0answers
159 views

What is the meaning of the coefficients of the Alekseev-Torossian associator

Drinfeld associators became a central object in mathematics and mathematical physics. They appear in deformation quantization, quantum groups, in the proof of the formality of the little disks operad, ...
10
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0answers
244 views

Computing the equivariant cohomology class of a Białynicki-Birula cell

One of my current research interests is Hessenberg varieties. Briefly, if $m_1\le m_2\le \cdots \le m_{n-1}$ is a weakly increasing sequence of positive integers such that $i\le m_i\le n$ for all $i$, ...
8
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146 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^\...
8
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0answers
335 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 $1$ ...
8
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0answers
121 views

Continuous analogues of Schützenberger promotion

Has anyone studied continuous analogues of Schützenberger promotion, and in particular, a flow on (a suitable subset of) the order polytope of a poset? Here’s what I have in mind: Given a poset $P$, ...
8
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0answers
192 views

q-analog of $(d/dx) \binom{x}{k}$?

It is not hard to find easy formulas for the derivative of the function $\binom{x}{k}$, for instance it is not too hard to see (for $k$ an integer) that $\frac{d}{dx} \binom{x}{k} = \sum_{i=1}^k \...
8
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0answers
420 views

A relation between intersection and product on Boolean interval of finite groups

Let $[H,G]$ be a Boolean interval of finite groups (i.e. the lattice of intermediate subgroups $H \subseteq K \subseteq G$, is Boolean). For any element $K \in [H,G]$, let $K^{\complement}$ be its ...
7
votes
0answers
213 views

Characterizing $n$-exceptions of the ring of symmetric polynomials

(Also in Mathematics Stack Exchange: https://math.stackexchange.com/questions/2528000/characterizing-n-exceptions-on-the-ring-of-symmetric-polynomials) We say that an homogeneous symmetric polynomial ...
6
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0answers
71 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 ...
6
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0answers
171 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 ...
6
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0answers
164 views

Is there are good algebraic model of random n-hypergraphs?

Suppose $F$ is a finite field and $-1$ is a square in $F$. Let $E$ be the binary relation on $F$ where $(a,b) \in E$ iff $a - b$ is a square. Then $(F,E)$ is called a Paley graph. Paley graphs are ...
6
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0answers
296 views

Coefficients in expansion of a classical symmetric polynomial

If we expand \begin{equation} P_3(x_1,\ldots, x_n):=\Pi_{1\leq i<j<k\leq n} (x_i+x_j+x_k), \end{equation} then \begin{equation} P_3 = \sum_\alpha \sum_{\mathcal{O}(\alpha)} c_\alpha x_1^{\...
5
votes
0answers
182 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. ...
5
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0answers
66 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 \...
5
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0answers
61 views

Embedding the Mészáros subdivision algebra in an Orlik-Terao localization

The following is an open question (Question 4.1) from my paper $t$-Unique Reductions for Mészáros's Subdivision Algebra (published version in SIGMA 2018, and slightly updated preprint version with ...
5
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0answers
212 views

Tabloid Construction of Permutation Representation of Hyperoctahedral Group

For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\...
5
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0answers
815 views

Internal tensor product of strict polynomial functors: is there a more explicit definition?

In the paper Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors, arXiv:1203.0311v4, Krause defines something that he calls an "internal tensor product" on the category of ...
5
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0answers
137 views

Is the Euler characteristic of a subfactor planar algebra, nonzero?

Let $\mathcal{P}$ be an irreducible subfactor planar algebra and $\mu$ the Möbius function of its biprojection lattice $[e_1,id]$. Then the Euler characteristic of $\mathcal{P}$ is defined as follows: ...
5
votes
0answers
456 views

Identities satisfied by the image of the Young symmetrizer

Consider a partition $\lambda=(r_1,\ldots,r_k)$ of an integer $n$ and the corresponding Young diagram with rows of length $r_1,\ldots,r_k$ (hence ordered in non-increasing order). Counting the column ...
5
votes
0answers
146 views

Counting square zero forms over finite fields

Let $p$ be an odd prime and let $R=\Lambda_{\mathbb{F}_p}[x_1,\dots,x_n]$ be the exterior algebra on $n$ generators over the finite field with $p$ elements. This is a graded-commutative ring. Is ...
5
votes
0answers
209 views

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 $A$...
5
votes
0answers
230 views

Counting perfect matchings with integrals

Has anyone used the Joni-Rota-Godsil integral formula (see details below) to count perfect matchings of square-grid graphs, Aztec diamond graphs, hexagon-honeycomb graphs, etc.? (Or even just to ...
5
votes
0answers
344 views

Littlewood-Richardson rule for Schubert polynomials

What is the current state of the problem of finding a combinatorial rule for multiplying two Schubert polynomials? Is the problem still open?
4
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0answers
135 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,...
4
votes
0answers
105 views

A combinatorial proof of an identity of partitions (Macdonald I.5)

This is a statement from Symmetric Functions and Hall Polynomials by Macdonald: $\sum_{x\in \lambda} (h(x)^2-c(x)^2)=|\lambda|^2$ where $\lambda$ denotes a partition or a Young diagram, and for each ...
4
votes
0answers
228 views

Generalization of a theorem of Øystein Ore in group theory: the infinite case

This post is the infinite version of this one, and is motivated by an exchange with Carmela Musella and Maria De Falco. We are interested in relative versions of the following Ore's theorem and ...
4
votes
0answers
171 views

Local structure of non-normal toric varieties---possible mistake in "Discriminants, Resultants and Multidimensional Determinants"

I believe I may have a counterexample to Theorem 5.3.1 on page 179 from the book book Discriminants, Resultants and Multidimensional Determinants by Gel'fand, Kapranov, and Zelevinsky. To summarize ...
4
votes
0answers
88 views

Rationality of power series whose coefficients are the ranks of a sequence of matrices

Recently, I stumbled several times about the problem to decide whether a certain formal power series $$ f = \sum_{n=0}^\infty d_n T^n \in \mathbb{Q}[\![T]\!]$$ is actually a rational function, where ...
4
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0answers
779 views

Categorifying the Cauchy kernel as a filtration of $\operatorname*{Sym}\left( F\otimes G\right) $ over any commutative ring

Question 1 (short version). Let $R$ be a commutative ring with unity. Let $F$ and $G$ be two $R$-modules. Let $n\in\mathbb{N}$. Is it true that the $n$-th symmetric power $\operatorname*{Sym}\...
4
votes
0answers
156 views

Chern character of Schubert structure sheaf

Let $X_\lambda \subset Gr = Gr(k,n)$ be a Schubert variety in the Grassmannian and $\mathrm{ch} : K_0(Gr) \otimes \mathbb{Q} \to A^\bullet(Gr) \otimes \mathbb{Q}$ the Chern character isomorphism. Is ...
4
votes
0answers
332 views

Can the Littlewood-Richardson cone be used for combinatorial optimization?

The Littlewood-Richardson cone $LR_{n, k}$ consists of all $k-$tuples $(a_1, a_2, \dots, a_k)$ of real $n-$vectors with monotonically decreasing entries such that there exist $k$ $n \times n-$...
3
votes
0answers
101 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 ...
3
votes
0answers
98 views

Can equality of chromatic symmetric functions of two trees be checked in polynomial time?

Stanley defined chromatic symmetric functions (CSF) in 1995 (Advances in Math) where he conjectured that trees can be distinguished by their CSF. However, tree isomorphism is decidable in P (...
3
votes
0answers
72 views

Subalgebras of a polynomial ring carved out by (families of) coefficient equalities

Let $\mathbf{k}$ be a field, and let $P=\mathbf{k}\left[ x_{1},x_{2} ,\ldots,x_{n}\right] $ be a polynomial ring over $\mathbf{k}$ in $n$ variables $x_{1},x_{2},\ldots,x_{n}$. Alternatively, $P$ can ...
3
votes
0answers
116 views

Number of adjoint orbits containing a $(0,1)$-matrix

Motivated by this question, what can be said about the number $f(n)$ of adjoint orbits of $\mathrm{Mat}(n,\mathbb{C})$ (the ring of all $n\times n$ complex matrices) that contain a $(0,1)$-matrix? ...
3
votes
0answers
78 views

Decomposition of Schur modules over the orthogonal group

Let $V=\mathbb{R}^n$ and $O(n)$ the orthogonal group acting with its standard action on $V$. Now for any partition $\lambda$ we have the Schur module $S_\lambda V$ which is a representation of $O(n)$. ...
3
votes
0answers
747 views

Frobenius formula

I know two formulas by the name of Frobenius. The first one computes the number $$\mathcal{N}(G;C_1,\dotsc,C_k):=|\{(c_1,\dotsc,c_k)\in C_1 \times \cdots \times C_k\:|\:c_1\cdots c_k=1\}|,$$ where $...
3
votes
0answers
104 views

Shifted schur function and holonomic

Now let us denote by $\Lambda^{*}(n)$ the algebra of polynomials in $x_{1},\ldots,x_{n}$ that become symmetric in new variables $$ x_{i}'=x_{i}-i+c, \ i \in 1,\ldots,n.$$ Here c is a arbitrary fixed ...
3
votes
0answers
688 views

Puzzle in 3D grid with black and white boxes, related to shelling

Consider a $n$ by $n$ by $n$ grid represented by the set of $3$-uples $S=\{1,2,\dots, n\}^3$. A line (resp. slice) of $S$ is a subset of cardinal $n$ (resp. $n^2$) where two components (resp. one ...
3
votes
0answers
478 views

Union of the conjugates of maximal subgroups

This post is a generalization of Union of the conjugates of a proper subgroup. Consider an interval $[H,G]$ in the subgroup lattice of the finite group $G$, with $H \neq G$ and such that: (1) $ \...
3
votes
0answers
194 views

Hook-content polynomial

Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\...