All Questions
7
questions
2
votes
0
answers
63
views
When does multiplying by an involution increase the Bruhat Order in the Symmetric Group?
Let $w \in \mathrm{Sym}(n)$ for some positive integer $n$. Let $r$ be an involution in $\mathrm{Sym}(n)$, and write it as the product of disjoint transpositions like so: $$r = \prod_{i=1}^k (a_i,b_i) $...
- 317
0
votes
0
answers
63
views
Coxeter graph of the group $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$
I am reading the first chapter of Combinatorics of Coxeter Groups by A.Björner and F.Brenti. In the first example they say that the graph with $n$ isolated vertices (no edges) is the Coxeter graph of ...
- 621
1
vote
0
answers
70
views
Counting inversions of random elements in coxeter groups
I am trying to find a general interperetation to the following facts (pls be patient to read it).
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of ...
0
votes
1
answer
315
views
Why $(1+x)(1+x+x^2)\cdots(1+x+x^2+\cdots+x^n)$ gives the Poincare series?
I am looking for an explanation of the fact that the polynomial $$(1+x)(1+x+x^2)...(1+x+x^2+...+x^n)$$ with Mahonian numbers gives the Poincare series for the symmetric group $S_{n+1}$ considered as a ...
3
votes
1
answer
150
views
Reduced word of a transposition
Take the group $S_n$ of permutations of $n$ points and as generator set take $S = \{s_1 := (1,2), s_2 := (2,3),\ldots, s_{n-1} := (n-1,n)\}$ the set of Coxeter generators. Let $t = (i,j) \in S_n$ be a ...
- 6,409
10
votes
1
answer
249
views
Description of flipping tableau for inversions in reduced decompositions of permutations
Short version: Is there a graphical description of the possible orders in which inversions can appear in a reduced decomposition of a permutation?
Something akin to the definition of standard Young ...
- 55.9k
15
votes
2
answers
1k
views
Finite/Infinite Coxeter Groups
In the same contest as this we got the following problem:
We are given a language with only three letters letters $A,B,C$. Two words are equivalent if they can be transformed from one another using ...
- 23.5k