Bruhat intervals, polyhedral cones and Kazhdan-Lusztig-Stanley polynomials. (English) Zbl 0809.20029
Kazhdan and Lusztig have associated a polynomial to each Bruhat interval in a Coxeter group. Similarly, R. Stanley has associated a polynomial to an Eulerian lattice. This paper marks the beginning of an attempt to study positivity of these polynomials in general by relating them to the representation theory of finite-dimensional algebras which can (conjecturally) be constructed form a certain labeling (by elements of a vector space) of the Hasse diagram of the associated poset. In this paper the author gives the definition and basic properties of the relevant labeling of Bruhat intervals and face lattices of polyhedral cones. The author also gives a common reformulation of the definition of the Kazhdan-Lusztig and Stanley polynomials in terms of convexity properties of the labels.
Reviewer: Chen Zhijie (Shanghai)
MSC:
| 20G05 | Representation theory for linear algebraic groups |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 06A07 | Combinatorics of partially ordered sets |
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
Keywords:
Kazhdan-Lusztig polynomials; Bruhat interval; Coxeter group; Eulerian lattice; positivity; finite-dimensional algebras; Hasse diagram; poset; labeling of Bruhat intervals; face lattices; polyhedral cones; Stanley polynomialsReferences:
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