The nil Hecke ring and Deodhar’s conjecture on Bruhat intervals. (English) Zbl 0813.20043
The author uses the properties of the “nil Hecke ring” proved by B. Kostant and M. Kumar [Adv. Math. 62, 187-237 (1986; Zbl 0641.17008)] to prove the following: Theorem. Let \((W,R)\) be a Coxeter system and \(\ell : W \to \mathbb{N}\) be the associated length function. Denote the Bruhat order on \(W\) by \(\leq\). Then for any \(v \leq y \leq w\) one has \(\#\{t \in T\mid v\leq ty \leq w\} \geq \ell(w) - \ell(v)\) where \(T\) denotes the set of reflections of \(W\), i.e. the conjugates in \(W\) of the simple reflections.
Reviewer: A.Khammash (Makkah)
MSC:
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 20F05 | Generators, relations, and presentations of groups |
Citations:
Zbl 0641.17008References:
| [1] | Carrell, J.B.: The Bruhat graph of a Coxeter group, a conjecture of Deodhar and rational smoothness of Schubert varieties. (Preprint) · Zbl 0818.14020 |
| [2] | Deodhar, V.: Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function. Invent. Math.39, 187-198 (1977) · doi:10.1007/BF01390109 |
| [3] | Deodhar, V.: On Bruhat orderings and Verma modules. INSA Publ. Sci. Acad. Medals Lectures, 69-75 (1978) |
| [4] | Deodhar, V.: On the root system of a Coxeter group. Commun. Algebra10, 611-630 (1982) · Zbl 0491.20032 · doi:10.1080/00927878208822738 |
| [5] | Deodhar, V.: Local Poincaré duality and non-singularity of Schubert varieties. Commun. Algebra13, 1379-1388 (1986) · Zbl 0579.14046 · doi:10.1080/00927878508823227 |
| [6] | Dyer, M.: Hecke algebras and shellings of Bruhat intervals II; twisted Bruhat orders. (Contemp. Math., volume on Kazhdan-Lusztig theory and related topics) (to appear) · Zbl 0833.20048 |
| [7] | Kostant, B., Kumar, S.: The nil Hecke ring and cohomology ofG/P for a Kac-Moody groupG. Adv. Math.62, 187-237 (1986) · Zbl 0641.17008 · doi:10.1016/0001-8708(86)90101-5 |
| [8] | Lakshmibai, V., Seshadri, C.S.: Singular locus of a Schubert variety. Bull. Am. Math. Soc., New Ser.11, 363-366 (1984) · Zbl 0549.14016 · doi:10.1090/S0273-0979-1984-15309-6 |
| [9] | Steinberg, R.: Endomorphisms of linear algebraic groups. Mem. Am. Math. Soc.80, 1-108 (1968) · Zbl 0164.02902 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
