The Enumeration of Fully Commutative Elements of Coxeter Groups
نویسنده
چکیده
A Coxeter group element w is fully commutative if any reduced expression for w can be obtained from any other via the interchange of commuting generators. For example, in the symmetric group of degree n, the number of fully commutative elements is the nth Catalan number. The Coxeter groups with finitely many fully commutative elements can be arranged into seven infinite families An , Bn , Dn , En , Fn , Hn and I2(m). For each family, we provide explicit generating functions for the number of fully commutative elements and the number of fully commutative involutions; in each case, the generating function is algebraic.
منابع مشابه
Characterization of cyclically fully commutative elements in finite and affine Coxeter groups
An element of a Coxeter group is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied by Boothby et al. In particular the authors precisely identified the Coxeter groups ...
متن کاملOn the cyclically fully commutative elements of Coxeter groups
Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W . We say that w is cyclically fully commutative (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced express...
متن کاملOn the Fully Commutative Elements of Coxeter Groups
Let W be a Coxeter group. We define an element w ~ W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose membe...
متن کاملFully commutative elements of type D and homogeneous representations of KLR-algebras
In this paper, we decompose the set of fully commutative elements into natural subsets when the Coxeter group is of type Dn, and study combinatorics of these subsets, revealing hidden structures. (We do not consider type An first, since a similar decomposition for type An is trivial.) As an application, we classify and enumerate the homogeneous representations of the Khovanov–Lauda– Rouquier al...
متن کاملFully Commutative Elements and Kazhdan–lusztig Cells in the Finite and Affine Coxeter Groups
The main goal of the paper is to show that the fully commutative elements in the affine Coxeter group e Cn form a union of two-sided cells. Then we completely answer the question of when the fully commutative elements of W form or do not form a union of two-sided cells in the case where W is either a finite or an affine Coxeter group. Let W be a Coxeter group with S the distinguished generator ...
متن کامل