Generalized Burnside algebra of type B_{n}
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Date
2020-06-30
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Ankara Üniversitesi Fen Fakültesi
Abstract
In this paper, we firstly give an alternative method to determine the size of
C
(
S
n
)
which is the set of elements of type
S
n
in a finite Coxeter system
(
W
n
,
S
n
)
of type
B
n
. We also show that all cuspidal classes of
W
n
are actually the conjugate classes
K
λ
for every
λ
∈
D
P
+
(
n
)
. We then define the generalized Burnside algebra
H
B
(
W
n
)
for
W
n
and construct a surjective algebra morphism between
H
B
(
W
n
)
and Mantaci-Reutenauer algebra
M
R
(
W
n
)
. We obtain a set of orthogonal primitive idempotents
e
λ
,
λ
∈
D
P
(
n
)
of
H
B
(
W
n
)
, that is, all the characteristic class functions of
W
n
. Finally, we give an effective formula to compute the number of elements of all the conjugate classes
K
λ
,
λ
∈
D
P
(
n
)
of
W
n
.
Description
Keywords
Cuspidal Class, Mantaci-Reutenauer Algebra, Burnside Algebra