All Questions
Tagged with symmetric-groups coxeter-groups
25 questions
6
votes
3
answers
434
views
What is known about finite dimensional modules over the nilCoxeter algebra?
Recall that the nilCoxeter algebra $\mathcal{N}_W$ for a Coxeter group $W$ is given by the $\mathbf{k}$-basis $x_w$ for each $w\in W$ and multiplication $x_ux_v=x_{uv}$ if $\ell(uv)=\ell(u)+\ell(v)$ ...
- 1,066
5
votes
1
answer
212
views
What is the effect of tensoring with the sign representation on irreducible modules for a Type D Weyl group?
Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...
- 2,160
2
votes
1
answer
145
views
When are these irreducible complex representations for the Type D Weyl group self-dual?
Given an integer $n \geq 4$, consider the Weyl groups $W(B_n)$ and $W(D_n)$ of types $B_n$ and $D_n$, respectively, and consider their representation theory over the field of complex numbers.
The Weyl ...
- 2,160
1
vote
1
answer
113
views
Minimal dominant permutation in weak order
Consider $S_\infty$ as a Coxeter group with Coxeter generators the adjacent transpositions $s_i$, $i\geq 1$. We view elements of $S_\infty$ as functions $u:\mathbb{N}\to\mathbb{N}$. Recall the Lehmer ...
- 2,168
8
votes
2
answers
282
views
One element commutation classes of reduced decompositions of the longest element of the Weyl group
For the symmetric group on $n$ objects $S_n$ the question of how to write its longest element $w_0$ as a reduced decomposition is an important combinatorical problem. As example, in this question the ...
2
votes
0
answers
283
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 ...
- 232
7
votes
1
answer
591
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. ...
- 33.8k
4
votes
0
answers
145
views
Words that give rise to an enumeration of elements of the symmetric group
Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration ...
- 1,066
4
votes
1
answer
296
views
Number of paths in the Bruhat order in the symmetric group
Let $\mathbb{S}_m$ the symmetric group on $m$ letters. Let $v\in\mathbb{S}_m$, and consider paths in the Bruhat order like this: $1\lessdot v_1\lessdot\cdots\lessdot v$, where $\lessdot$ means the ...
- 1,066
2
votes
1
answer
257
views
Recurrence relation for number of reduced words of longest element in $S_n$
Is there any recurrence relation known for the number of reduced words of the longest element in $S_n$ (not commutation classes)?
Edit: Sorry for unaccepting the answer, but I realized that I really ...
- 1,813
2
votes
0
answers
47
views
Carter Payne homomorphisms and reduced expressions
Let $G$ be an algebraic group and $W$ denote the underlying affine Weyl group. I will label representations of the principal block of $G$ by their alcoves, which in turn I label by the corresponding ...
- 1,413
1
vote
0
answers
147
views
A certain kind of permutations and transport of Bruhat chains under conjugation
Let $(W,S)$ be a finite Coxeter system. Let us consider the following situation:
Let $v_1,v_2,w\in W$ such that $v_1=wv_2w^{-1}$. Let $s_{\beta_r}\ldots s_{\beta_1}$ be a reduced expression of $v_2$. ...
- 1,066
1
vote
0
answers
176
views
Relation between groups $A_n$, $B_n$, $D_n$ and $S_n$ or inversions of random elements in Coxeter groups
First of of all I'm trying to find a general interpretation to the following facts below.
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian ...
1
vote
0
answers
130
views
Word length norm in the symmetric group $\mathfrak{S}_r$
Consider on the symmetric group $\mathfrak{S}_r$ the generating system $\{\tau_i;\,1\le i\le r-1\}$ with $\tau_i = \langle i,i+1\rangle$ and the corresponding word length norm $N$. Now let $\tau\in\...
- 1,623
4
votes
0
answers
98
views
Ref. request: Enumerating elements of Bruhat cells
Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then
$$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$
where we embed the ...
- 809
8
votes
1
answer
203
views
Reference request: Coxeter length and irreducible characters
Let $S_n$ be the symmetric group on $\{1,2,\ldots, n\}$ and $\ell$ the Coxeter length on $S_n$. There is a well-known formula to compute this length, namely for a $\pi \in S_n$ we have
$$\ell(\pi) = |\...
- 809
12
votes
1
answer
385
views
Question on a reduction in Kirillov's paper on positivity of divided difference operators
As the title says, my question is on a specific argument in Kirillov - Skew divided difference operators and Schubert polynomials (journal, MSN) on positivity of divided difference operators. I recall ...
- 1,066
2
votes
0
answers
81
views
Characterization of permutations which have at most one successor in the covering relation of the weak Bruhat order
Let $W$ be the symmetric group on $n+1$ letters. Let $\ell$ be the length function on $W$.
As the title says, can we characterize all $v\in W$ such that there exists a $w\in W$ such that for all ...
- 1,066
1
vote
2
answers
259
views
A question about set of inversion
Let $w \in S_n$ and $inv(w) = \{(i,j): i,j \in \{1,\ldots,n\}, i<j, w(i)>w(j)\}$ the inversion set of $w$. Let ${\bf i}=(i_1,\ldots,i_m)$ be a sequence such that $s_{i_1}\cdots s_{i_m}$ is a ...
- 6,201
6
votes
1
answer
301
views
Reference request: Reduced reflection length in Coxeter groups
I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. ...
- 809
0
votes
1
answer
232
views
Number of Boolean algebra subintervals in weak order of $S_n$
I'm wondering if anybody has an easy way to compute the number of subintervals in weak order of $S_n$ (considered as a Coxeter group of type $A_{n-1}$) that are isomorphic to Boolean algebras. I know $...
- 2,168
3
votes
0
answers
414
views
Kazhdan-Lusztig basis of the symmetric group algebra $\mathbb{K}S_n$
Let $S_n$ be the symmetric group on the set $\{1,2,\ldots,n\}$ and $\mathbb{K}$ a field. Denote by $\mathbb{K}S_n$ the symmetric group algebra.
What's Kazhdan-Lusztig basis of $\mathbb{K}S_n$? I ...
- 331
4
votes
3
answers
947
views
Maximal pairwise distance between $k$ permutations
How can k permutations on n-set be arranged to maximize minimal pairwise Kendall tau distance (i.e. number of discordant pairs) between them?
For two permutations this is obviously when the second ...
10
votes
4
answers
940
views
Is there a non-explicit characterization of the Specht modules?
It is a basic fact about the symmetric group $S_n$ that its irreducible representations are indexed by partitions of $n$.
My question is, can the association between partitions and irreps be ...
- 258
11
votes
2
answers
2k
views
Algorithm for reducing words in a Coxeter group
Let $W$ be a Coxeter group with set of simple reflections $S$. Suppose that I have chosen a preferred reduced decomposition for every element of $W$. Given an arbitrary word in the alphabet $S$, is ...
- 213