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Representations of Hecke Algebras of finite groups with BN-Pairs of classical type Hoefsmit, Peter Norbert
Abstract
Let G be a finite group with BN-pair and Coxeter system (W, R). Let A be the generic ring corresponding to (W, R) in the sense of Tits, defined over the polynomial ring D = Q[u[sub r], r ε R]. Let k be any field of characteristic zero. For the homomorphism φ : D —> k defined by φ(u[sub r]) = q[sub r], q[sub r] the index parameters of G, the specialzed algebra Aφ,[sub k] is isomorphic to the Hecke algebra H[sub k](G, B) of G with respect to a Borel subgroup B of G, while for the specialization defined by φ(u[sub r]) = 1, r ε R, Aφ,[sub k] is isomorphic to the group algebra kW. As G the Hecke algebra H[sub k](G, B) affords the induced representation 1[sup G sub B], the G irreducible representations of G appearing in 1[sup G sub B] can be obtained from the representations of H[sub k](G, B). In this thesis, we obtain all the irreducible representations, defined over the quotient field of D, of the generic ring corresponding to a Coxeter system of classical type. The method employed involves a generalization of Young's construction of the semi-normal matrix representations of the symmetric group. We also obtain an explicit formula for the generic degree of these representations in terms of the hook lengths of Young diagrams. Thus the degrees of all the irreducible constituents of 1[sup G sub B] are obtained for the families of Chevalley groups A[sub l](q), B[sub l](q), A[sup 1 sub 2l](q²), A[sup 1 sub 2l-1](q²), D[sup 1 sub l](q²) and for D[sub l](q), l odd. Also, most of the degrees are obtained for D[sub l](q), l even.
Item Metadata
Title |
Representations of Hecke Algebras of finite groups with BN-Pairs of classical type
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
1974
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Description |
Let G be a finite group with BN-pair and Coxeter system
(W, R). Let A be the generic ring corresponding to (W, R) in the sense
of Tits, defined over the polynomial ring D = Q[u[sub r], r ε R]. Let k be
any field of characteristic zero. For the homomorphism φ : D —> k
defined by φ(u[sub r]) = q[sub r], q[sub r] the index parameters of G, the specialzed
algebra Aφ,[sub k] is isomorphic to the Hecke algebra H[sub k](G, B) of G with
respect to a Borel subgroup B of G, while for the specialization defined
by φ(u[sub r]) = 1, r ε R, Aφ,[sub k] is isomorphic to the group algebra kW. As
G
the Hecke algebra H[sub k](G, B) affords the induced representation 1[sup G sub B], the
G
irreducible representations of G appearing in 1[sup G sub B] can be obtained from the representations of H[sub k](G, B).
In this thesis, we obtain all the irreducible representations, defined over the quotient field of D, of the generic ring corresponding to a Coxeter system of classical type. The method employed involves a generalization of Young's construction of the semi-normal matrix representations
of the symmetric group.
We also obtain an explicit formula for the generic degree of these representations in terms of the hook lengths of Young diagrams. Thus the degrees of all the irreducible constituents of 1[sup G sub B] are obtained for the
families of Chevalley groups
A[sub l](q),
B[sub l](q),
A[sup 1 sub 2l](q²),
A[sup 1 sub 2l-1](q²),
D[sup 1 sub l](q²)
and for D[sub l](q), l odd. Also, most of the degrees are obtained for D[sub l](q),
l even.
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Genre | |
Type | |
Language |
eng
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Date Available |
2010-01-25
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Provider |
Vancouver : University of British Columbia Library
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Rights |
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.
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DOI |
10.14288/1.0080112
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Campus | |
Scholarly Level |
Graduate
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Aggregated Source Repository |
DSpace
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Rights
For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.