scholarly journals Affine Weyl Groups as Infinite Permutations

10.37236/1356 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Henrik Eriksson ◽  
Kimmo Eriksson
Keyword(s):  
Poincaré Series ◽  
Weak Order ◽  
Length Function ◽  
Weyl Groups ◽  
Unified Theory ◽  
Affine Weyl Groups ◽  
Poincare Series ◽  
Permutation Models ◽  
Refined Version

We present a unified theory for permutation models of all the infinite families of finite and affine Weyl groups, including interpretations of the length function and the weak order. We also give new combinatorial proofs of Bott's formula (in the refined version of Macdonald) for the Poincaré series of these affine Weyl groups.

scholarly journals An Order on Circular Permutations

10.37236/9982 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Antoine Abram ◽  
Nathan Chapelier-Laget ◽  
Christophe Reutenauer
Keyword(s):  
Symmetric Group ◽  
Weak Order ◽  
Rank Function ◽  
Weyl Groups ◽  
Eulerian Numbers ◽  
Affine Weyl Groups ◽  
Special Formula ◽  
Circular Permutations ◽  
Semidistributive Lattice ◽  
Young’S Lattice

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

scholarly journals The distribution of descents and length in a Coxeter group

10.37236/1219 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Victor Reiner
Keyword(s):  
Rational Function ◽  
Coxeter Group ◽  
Generating Functions ◽  
Length Function ◽  
Weyl Groups ◽  
Coxeter System ◽  
Affine Weyl Groups ◽  
System L

We give a method for computing the $q$-Eulerian distribution $$ W(t,q)=\sum_{w \in W} t^{{\rm des}(w)} q^{l(w)} $$ as a rational function in $t$ and $q$, where $(W,S)$ is an arbitrary Coxeter system, $l(w)$ is the length function in $W$, and ${\rm des}(w)$ is the number of simple reflections $s \in S$ for which $l(ws) < l(w)$. Using this we compute generating functions encompassing the $q$-Eulerian distributions of the classical infinite families of finite and affine Weyl groups.

scholarly journals Solution of tetrahedron equation and cluster algebras

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
P. Gavrylenko ◽  
M. Semenyakin ◽  
Y. Zenkevich
Keyword(s):  
Integrable System ◽  
Cluster Algebra ◽  
Newton Polygon ◽  
Cluster Algebras ◽  
Weyl Groups ◽  
Newton Polygons ◽  
Tetrahedron Equation ◽  
Affine Weyl Groups ◽  
New Formalism ◽  
Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

scholarly journals Poincaré series, 3d gravity and averages of rational CFT

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Viraj Meruliya ◽  
Sunil Mukhi ◽  
Palash Singh
Keyword(s):  
Poincaré Series ◽  
Simons Theory ◽  
Partition Functions ◽  
Moody Algebra ◽  
Modular Invariant ◽  
Poincare Series ◽  
Single Genus ◽  
3D Gravity ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

Sign in / Sign up

Export Citation Format

Share Document