The representation algebra

Throughout this paper F is an algebraically closed field of characteristic p (≠ 0) and g is a finite group whose order is divisible by p. We define in the usual way an F-representation of g (or F G-representation) and its corresponding module. The isomorphism class of the, F G-representation module M is written {M} or, where no confusion arises, M. A (G) denotes the F-representation algebra of G over the complex field G (as defined on pages 73 and 82 of [6]).

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TheL ∞-representation algebra of an idempotent topological semigroup

Semigroup Forum ◽

10.1007/bf02573541 ◽

1993 ◽

Vol 46 (1) ◽

pp. 32-36 ◽

Cited By ~ 1

Author(s):

J. W. Baker ◽

M. Lashkarizadeh-Bami

Keyword(s):

Topological Semigroup ◽

Representation Algebra

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The Coarse Structure of the Representation Algebra of a Finite Monoid

Journal of Discrete Mathematics ◽

10.1155/2014/529804 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Mary Schaps

Keyword(s):

Fine Structure ◽

Finite Monoid ◽

Coarse Structure ◽

Primitive Idempotents ◽

Complete Set ◽

One To One ◽

Representation Algebra

Let M be a monoid, and let L be a commutative idempotent submonoid. We show that we can find a complete set of orthogonal idempotents L^0 of the monoid algebra A of M such that there is a basis of A adapted to this set of idempotents which is in one-to-one correspondence with elements of the monoid. The basis graph describing the Peirce decomposition with respect to L^0 gives a coarse structure of the algebra, of which any complete set of primitive idempotents gives a refinement, and we give some criterion for this coarse structure to actually be a fine structure, which means that the nonzero elements of the monoid are in one-to-one correspondence with the vertices and arrows of the basis graph with respect to a set of primitive idempotents, with this basis graph being a canonical object.

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The identity of certain representation algebra decompositions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700045664 ◽

1970 ◽

Vol 3 (1) ◽

pp. 73-74

Author(s):

S. B. Conlon ◽

W. D. Wallis

Keyword(s):

Finite Group ◽

Tensor Product ◽

Linear Algebra ◽

Residue Field ◽

Complex Field ◽

Direct Sums ◽

Isomorphism Classes ◽

Direct Sum ◽

A Finite Group ◽

Representation Algebra

Let G be a finite group and F a complete local noetherian commutative ring with residue field of characteristic p # 0. Let A(G) denote the representation algebra of G with respect to F. This is a linear algebra over the complex field whose basis elements are the isomorphism-classes of indecomposable finitely generated FG-representation modules, with addition and multiplication induced by direct sum and tensor product respectively. The two authors have separately found decompositions of A(G) as direct sums of subalgebras. In this note we show that the decompositions in one case have a common refinement given in the other's paper.

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