Questions tagged [algebraic-combinatorics]
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263
questions
3
votes
1
answer
178
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 ...
- 33
1
vote
0
answers
127
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 ...
- 3,370
0
votes
0
answers
64
views
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?
- 320
7
votes
0
answers
85
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$ ...
- 1,313
4
votes
1
answer
235
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. ...
- 215
5
votes
0
answers
123
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 ...
- 678
7
votes
0
answers
425
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 ...
- 106
4
votes
0
answers
178
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 ...
0
votes
0
answers
241
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 ...
- 5,981
2
votes
1
answer
216
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 ...
- 15.4k
14
votes
1
answer
515
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 $\...
- 440
7
votes
0
answers
103
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 ...
- 2,640
6
votes
1
answer
171
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 ...
- 2,640
8
votes
0
answers
303
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}...
- 1,070
9
votes
0
answers
226
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 ...
- 16.2k
8
votes
0
answers
213
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
$$ \langle p_\lambda,p_\mu\rangle =
\delta_{\lambda\mu}z_\...
- 47.9k
0
votes
0
answers
83
views
Addition theorem for Schur function in multivariable
Working with the following problem Expansion in Schur function of negative binomial exponent
I need to find the expansion of
$$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$
in terms of schur ...
- 675
3
votes
0
answers
131
views
Expansion in Schur function of negative binomial exponent
I want to know if there exist a known expansion or can be derived of the polynomial
$$ \prod_{i=1}^{m}\prod_{j= 1}^{n}(1-z(x_i + y_i))^{-w} \tag{*}$$
in terms of Schur function. That is asking for (*) ...
- 675
5
votes
1
answer
125
views
PBW basis for the quantized enveloping Lie algebra of $\mathfrak{g}_2$
I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig.
Alternatively I ...
3
votes
0
answers
66
views
Subrings of the ring of symmetric functions
While experimenting with symmetric functions, I noticed the following equality of subrings of the ring of symmetric functions:
$$\mathbb{Z}[(n-1)!\cdot p_n \ |\ n \ge 1] = \mathbb{Z}[n!\cdot h_n \ |\ ...
- 2,640
2
votes
3
answers
384
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 ...
- 511
15
votes
2
answers
576
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
$$ p_3 = x_1^3 + x_2^3 + \dots + x_n^3 , $$
$$ p_5 = ...
- 153
5
votes
1
answer
765
views
Expressing symmetric function in power-sum basis
I am trying to prove the following identity
\begin{equation}
\prod_{i=1}^{m}(1-x_{i}z)^{-u}\prod_{j=1}^{n}(1-y_{i}z)^{-v} \prod_{i=1}^{m}\prod_{j=1}^{n}(1-(x_i +y_j)z)^{-w}\\ = \sum_{\lambda, \mu}c_{\...
- 675
0
votes
0
answers
69
views
Algebraic ode of exponential generating series
Let $G(z)$ be a rational function. So if we have a series
$$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 ...
- 675
4
votes
2
answers
242
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 ...
- 5,450
2
votes
0
answers
132
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 ...
- 203
10
votes
1
answer
266
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 ...
- 2,640
17
votes
2
answers
766
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 \...
- 25.1k
0
votes
1
answer
243
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 ...
- 13
7
votes
0
answers
130
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 ...
- 171
8
votes
1
answer
249
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^\...
- 221
5
votes
1
answer
417
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. ...
- 32.2k
0
votes
0
answers
127
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 ...
7
votes
2
answers
441
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 $f$ ...
- 19k
1
vote
0
answers
74
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:...
- 227
15
votes
1
answer
679
views
a Vandermonde-type of determinants summed over permutations
Let $S_n$ be the symmetric group. Consider
$$D:=\sum_{\sigma\in S_n} \text{sgn}(\sigma)\cdot \det\begin{pmatrix}1 & a_{\sigma(1)}-0 & (a_{\sigma(1)}-0)^2 & \cdots & (a_{\sigma(1)}-0)^{...
- 151
9
votes
0
answers
469
views
Two majs for standard Young tableaux?
Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...
- 21.3k
4
votes
1
answer
228
views
proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions"
I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...
- 41
3
votes
0
answers
193
views
Does every finite lattice embed into a finite Eulerian lattice?
A finite Boolean lattice is a lattice isomorphic to the subset lattice of a finite set. Every Boolean lattice is Eulerian, namely, a graded lattice $L$ such that $\mu(a,b) = (-1)^{|b|-|a|}$ for all $a,...
- 25.1k
5
votes
0
answers
92
views
Form on symmetric functions and their q,t- analogues
[Notations are as in Macdonald's Symmetric Functions and Hall Polynomials]
The space of symmetric functions $\Lambda_{\mathbb{Q}}$ has a bilinear form defined by $ (p_\lambda, p_\mu)= z_\lambda \...
- 678
2
votes
0
answers
84
views
Double Schur function expansion
In literature, I have seen the weighted Hurwitz number $N_{g,n}(d_1 , d_2 \ldots , d_n)$ which are symmetric number and they can be written as double Schur function expansion.
\begin{align} \label{eq:...
- 675
9
votes
2
answers
491
views
Characterization of the family of simple groups PSL(2,q) by tensor multiplicity
Let $G$ be a finite group and $(\chi_i)$ its irreducible characters. Then $\forall i,j,k, \exists!n_{i,j}^k \in \mathbb{N}_{\ge 0}$ such that $$\chi_i \chi_j = \sum_k n_{i,j}^k \chi_k.$$
Let the ...
- 25.1k
3
votes
1
answer
218
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 ...
- 438
6
votes
0
answers
184
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 ...
- 678
3
votes
3
answers
401
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 ...
- 237
6
votes
1
answer
273
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 ...
- 32.2k
3
votes
0
answers
109
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 ...
- 1,036
9
votes
1
answer
229
views
Is there a 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 ...
- 1,599
3
votes
0
answers
112
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 (...
- 31
14
votes
2
answers
826
views
Do you know an elegant proof for this expression for a Schur function?
I know that the identity
$$
s_\mu = \sum_{\mu-\lambda \text{ is a horizontal strip}} \;\sum_{\alpha\vdash|\lambda|} \frac{\chi^\lambda_\alpha}{z_\alpha} \prod_i(p_i-1)^{a_i}
$$
holds.
Here $\alpha=1^{...
- 5,504