Questions tagged [algebraic-combinatorics]
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91
questions with no upvoted or accepted answers
27
votes
0
answers
876
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
votes
0
answers
868
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
votes
0
answers
381
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
votes
0
answers
375
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
votes
0
answers
452
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
votes
0
answers
187
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
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 ...
12
votes
0
answers
190
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 ...
12
votes
0
answers
702
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
$$S_u(x)S_v(x;y)=\sum_w{c_{...
10
votes
0
answers
204
views
Branching from GL(a+b) to GL(a) x GL(b)$ using Gel'fand-Cetlin patterns
If one iterates the multiplicity-free branching rule from $\operatorname{GL}(n)$ representations (finite-dim, over $\mathbb C$) to $\operatorname{GL}(n-1)$ all the way down to $\operatorname{GL}(0)$, ...
10
votes
0
answers
219
views
What is the meaning of the coefficients of the Alekseev-Torossian associator
Drinfeld associators became a central object in mathematics and mathematical physics. They appear in deformation quantization, quantum groups, in the proof of the formality of the little disks operad, ...
10
votes
0
answers
259
views
Computing the equivariant cohomology class of a Białynicki-Birula cell
One of my current research interests is Hessenberg varieties. Briefly, if $m_1\le m_2\le \cdots \le m_{n-1}$ is a weakly increasing sequence of positive integers such that $i\le m_i\le n$ for all $i$, ...
9
votes
0
answers
480
views
Two majs for standard Young tableaux?
Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...
8
votes
0
answers
128
views
Conceptual explanation for the gap in the spectrum of biregular trees
Ramanujan graphs are defined as $(q+1)$-regular graphs for which all nontrivial eigenvalues of the adjacency operator are contained in the interval
$$[-2\sqrt{q}, 2\sqrt{q}].$$
The reason for this ...
8
votes
0
answers
310
views
Why are these Littlewood-Richardson coefficients congruent to 1 mod 8?
Let $n\in{\mathbb N}$ and write $n=q_1+q_2+\dots+q_t$, where $q_1>q_2>\dots>q_t$ are powers of $2$. Let $\lambda_n$ be the partition with Frobenius symbol $(q_1-1,q_2-1,\dots,q_t-1;q_t,q_{t-1}...
8
votes
0
answers
225
views
Scalar products on symmetric functions behaving like the Macdonald scalar product
The Macdonald symmetric functions (or Macdonald polynomials)
$P_\lambda(x)$ are orthogonal with respect to the Macdonald scalar
product
$$ \langle p_\lambda,p_\mu\rangle =
\delta_{\lambda\mu}z_\...
8
votes
0
answers
150
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
votes
0
answers
241
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
votes
0
answers
434
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
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$ ...
7
votes
0
answers
437
views
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 ...
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 ...
7
votes
0
answers
237
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
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 ...
6
votes
0
answers
305
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^{\...
6
votes
0
answers
378
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?
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 ...
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 \...
5
votes
0
answers
75
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
votes
0
answers
1k
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 $...
5
votes
0
answers
254
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
votes
0
answers
998
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
votes
0
answers
139
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
0
answers
536
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
0
answers
149
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
0
answers
213
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
0
answers
240
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 ...
4
votes
0
answers
191
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 ...
4
votes
0
answers
131
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
0
answers
245
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
0
answers
208
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
0
answers
98
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
votes
0
answers
838
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
0
answers
160
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
0
answers
338
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
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 (*) ...
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 \ |\ ...
3
votes
0
answers
205
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,...
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 ...
3
votes
0
answers
119
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 (...