What is “distinguished cross section” (Solomon’s descent algebra proof)?

Question

I’m attempting to read the proof of Solomon’s descent algebra from A Mackey Formula in the Group Ring of Coxeter Groups though I know nothing about Coxeter groups and Mackey formulas. I’m interested just in the case of symmetric groups and have no idea how that statement of Theorem 1 is related to descents in permutations in any way.

Particularly, I’m stuck with the word “distinguished cross section.” Basic googling returns nothing so I would appreciate if anyone would help or point to some references. Also if you could help explain how descent algebra of symmetric group is the special case of that theorem. Thank you.

Answer 1

This appears to the the relevant definition/description:

For $w\in W$ let $\ell(w)$ denote the length of $w$ as a word in the elements of $S.$ Each $(W_J, W_K)$ double coset contains a unique element of minimal length [1, Chap. 4, Exercise 1.3]. Thus $W_J\big{\backslash} W \big{/} W_K$ has a distinguished cross section which we denote $X_{JK}.$ In case $J$ is empty we write $X_K = X_{JK}$ for the distinguished cross section of $W/W_K .$

From this (in particular, from the statement that "Each $(W_J, W_K)$ double coset contains a unique element of minimal length"), I would conclude that "cross section" means "set of representatives." That is, a cross section $X$ of $W_J\big{\backslash} W \big{/} W_K$ is a set of elements of $W$ such that for each double coset $W_J w W_K,$ there exists a unique $x\in X$ with $W_J w W_K = W_J x W_K.$

In particular, the distinguished cross section $X_{JK}$ is a set of representatives such that if $x\in X_{JK}$ and $x\neq w\in W$ with $W_J x W_K = W_J w W_K,$ then $\ell(x) < \ell(w).$ Remember that the length $\ell(x)$ is defined to be the minimal number $n$ such that $$ x = s_1^{\epsilon_1} s_2^{\epsilon_2}\dots s_n^{\epsilon_n} $$ with each $s_i\in S$ (the chosen generating set of $W$) and each $\epsilon_i = \pm1.$