Depth in classical Coxeter groups

Abstract

The depth statistic was defined by Petersen and Tenner for an element of an arbitrary Coxeter group in terms of factorizations of the element into a product of reflections. It can also be defined as the minimal cost, given certain prescribed edge weights, for a path in the Bruhat graph from the identity to an element. We present algorithms for calculating the depth of an element of a classical Coxeter group that yield simple formulas for this statistic. We use our algorithms to characterize elements having depth equal to length. These are the short-braid-avoiding elements. We also give a characterization of the elements for which the reflection length coincides with both depth and length. These are the boolean elements.

Affine Deligne-Lusztig Varieties and Quantum Bruhat Graph

“…i ρ, β ∨ i : x = s β1 • • • s β k , β i ∈ Φ + }. (8.1)Following[2], we say that the W -depth of x is realized by a reduced factorization of x if there existsan expression x = s β1 • • • s β k with β i ∈ Φ + , such that ℓ(x) = k i=1 ℓ(s βi ) and dp(x) = k i=1dp(s βi ).…”

“…[2, Proposition 6.6] If the W -depth of x is realized by a reduced factorization then dp(x) = 1 2 (ℓ(x) + ℓ red (x)); in particular, this equality holds whenever W is a classical finite Coxeter group. Relation between wt and dp.Lemma 8.4.…”

“…In fact, this is called depth in[3],[2] and[23]; we alter the terminology here to avoid any confusion with the notion of depth of a coweight defined earlier.…”

Bargain hunting in a Coxeter group

“…In [6], the authors considered the situation in which S n is generated by the set T = {(i j)} of all transpositions, and the cost function on transpositions is given by (1) $((i j)) = |j − i|, i.e., the cost of a transposition is equal to the distance between the points it transposes in the one-line notation of a permutation. They showed that for this function, the cost of a permutation is half of its total displacement:…”

The shallow permutations are the unlinked permutations

“…The shallow permutations have another surprising connection not previously noted in the literature. Given a permutation w, Bagno, Biagioli, Novick, and the last author [4] defined the reduced reflection length ℓ R (w) as the smallest integer q such that there exist i 1 , . .…”