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.
Depth in classical Coxeter groups / Biagioli R; Bagno E; Novick M; Woo A. - In: JOURNAL OF ALGEBRAIC COMBINATORICS. - ISSN 0925-9899. - STAMPA. - 44:(2016), pp. 645-676. [10.1007/s10801-016-0683-9]
Depth in classical Coxeter groups
Biagioli R;
2016
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.