scholarly journals Relations of the Weyl groups of extended affine root systems $A_1 ^{\left( {1,1} \right)} ,B_1 ^{\left( {1,1} \right)} ,C_1 ^{\left( {1,1} \right)} ,D_1 ^{\left( {1,1} \right)}$

1995 ◽  
Vol 71 (6) ◽  
pp. 123-124
Author(s):  
Tadayoshi Takebayashi
Keyword(s):  
Root Systems ◽  
Weyl Groups ◽  
Affine Root Systems

scholarly journals A Length Function for Weyl Groups of Extended Affine Root Systems of Type A1

2015 ◽  
Vol 22 (04) ◽  
pp. 621-638 ◽  
Author(s):  
Saeid Azam ◽  
Mohammad Nikouei
Keyword(s):  
Weyl Group ◽  
Root Systems ◽  
Type A ◽  
Length Function ◽  
Weyl Groups ◽  
Combinatorial Properties ◽  
Affine Root Systems

In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.

scholarly journals Zeta-functions of root systems and Poincaré polynomials of Weyl groups

2020 ◽  
Vol 72 (1) ◽  
pp. 87-126
Author(s):  
Yasushi Komori ◽  
Kohji Matsumoto ◽  
Hirofumi Tsumura
Keyword(s):  
Zeta Functions ◽  
Root Systems ◽  
Weyl Groups

INVERSIONS IN CLASSICAL WEYL GROUPS

2007 ◽  
Vol 09 (01) ◽  
pp. 1-20
Author(s):  
KEQUAN DING ◽  
SIYE WU
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Root Systems ◽  
Flag Manifold ◽  
Length Function ◽  
Weyl Groups ◽  
Flag Manifolds ◽  
A Finite Group

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

ROOT SYSTEMS ARISING FROM AUTOMORPHISMS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250057 ◽  
Author(s):  
SAEID AZAM ◽  
MALIHE YOUSOSFZADEH
Keyword(s):  
Lie Algebras ◽  
Root Systems ◽  
Affine Lie Algebras ◽  
Combinatorial Approach ◽  
Reflection Algebras ◽  
Affine Root Systems ◽  
Outstanding Work ◽  
Affine Reflection

We study a combinatorial approach of producing new root systems from the old ones in the context of affine root systems and their new generalizations. The appearance of this approach in the literature goes back to the outstanding work of Kac in the realization of affine Kac–Moody Lie algebras. In recent years, this approach has been appeared in many other works, including the study of affinization of extended affine Lie algebras and invariant affine reflection algebras.

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