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For the symmetric group $S_n$, an inversion of a permutation $\pi∈S_n$ is a pair $1\leq i<j\leq n$ such that $\pi(i)>\pi(j)$. It is known that the length $\ell(\pi)$ of a permutation (i.e. the least number of simple transpositions needed to express $\pi$) coincide with its number of inversions. This is useful to compute $$\sum_{\pi\in S_n}q^{\ell(\pi)} = (n)_q!$$ What happens if I replace $S_n$ by an arbitrary finite Coxeter group? Is there still a notion for an inversion? What is the above sum in this case?

Edit Okay, to be able say that I have my homework done, here some thoughts on the problem: A general definition of the inversion number for Weyl groups seems to be the following: The inversion number of a word $w$ is its number of positive roots that are mapped to negative roots. This still equals its length $\ell(w)$.

Example: $B_n$ For this group this seems to be easy (since I can draw pictures in my head for what it does…): It is simply the group $S_n$ with one additional generator that flips let's say the first sign in a sequence.

Apart from permuting elements, we may flip zero to $n$ signs, where we have $n$ possibilities for the first sign, $n-1$ for the second etc. So for $B_n$, the required polynomial should be

$$\sum_{w\in B_n}q^{\ell(w)} = (1+nq + (n-1)q^2 + \cdots + q^n)(n)_q!$$

which coincides with counting the elements explicitly (at least, for $B_2$)

  • Is there a nicer way to express this number?
  • Is there a general systematic, e.g. how to obtain this from the presentation?
  • Is there a listing for other Coxeter groups?
Bubaya
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1 Answers1

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As you might be well aware, given a Coxeter group $W$, the Poincare polynomial is defined to be $$ W(q)=\sum_{n\geq 0}a_nq^n=\sum_{w\in W}q^{\ell(w)}, $$ where $a_n=|\{w\in W:\ell(w)=n\}|$.

One place where computations of the Poincare polynomials can be found is in the paper "A combinatorial derivation of the Poincare polynomials of the finite irreducible Coxeter groups" by Rudolf Winkel. It covers types $A_n, B_n, D_n$, and the other exceptional types.

Some other basic results about Poincare polynomials/series can be found in Sections 1.11 and 5.12 of Humphreys' Reflection Groups and Coxeter Groups.

Ben West
  • 12,754
  • I was not aware that this polynomial has a name. This faciliates searching. But how does one see that one new generator in type D contributes $(1+q^{n-1})(n)_q$? I Always compute $(1+q^{n+2})(n)_q$, which seems to be wrong. – Bubaya Jul 21 '17 at 17:46