All Questions
4
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
2
votes
1
answer
64
views
Does the Coxeter group $C(D_n)$ have any "proper reflection quotients" except $C(A_{n-1})$?
Here, a reflection quotient is a surjective homomorphism between Coxeter groups mapping reflections to reflections. A reflection quotient $C(\Delta) \to C(\Gamma)$ is proper if it is not injective and ...
- 4,786
3
votes
1
answer
88
views
Orbits of pairs of multi-indices under the diagonal action of the symmetric group
This question concerns a statement made on page 168 of
Dipper, R. and Donkin, S., 1991. Quantum GLn. Proceedings of the London Mathematical Society, 3(1), pp.165-211.
I have tried to include all ...
- 2,251
2
votes
1
answer
644
views
Coxeter length in the symmetric group equals number of inversions
Let $S_n$ be the symmetric group on the set $\{1,\dots, n\}$ and $\sigma\in S_n$. An inversion of $\sigma$ is a pair $(i,j)$ such that $1\leq i<j\leq n$ and $\sigma(i)>\sigma(j)$. Let $S_n$ be ...
- 6,323