Regular Representation

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 22 January 2010

Regular representation

If A is an algebra then Aop is the algebra A except with the opposite multiplication, i.e. Aop=aop|a∈Awitha1opa2op=a1a2op,for alla1a2∈A.

The left regular representation of A is the vector space A with A action given by left multiplication. Here A is serving as both an algebra and an A-module. It is often useful to distinguish the two roles of A and use the notation A→ for the A-module, i.e. A→ is the vector space A→=b→|b∈AwithA-actionab→=ab→,for alla∈A,b→∈A→.

Let A be an algebra and let A→ be the regular representation of A. Then EndAA→≅Aop. More precisely EndAA→=φb|b∈A, where φb is given by φba→=ab→,for alla→∈A→.

		Proof.
	Let φ∈EndAA→ and let b∈A be such that φ1→=b→. For all a→∈A→, φa→=φa·1→=aφ1→=ab→, and so φ=φb. Then EndAA→≅Aop since φb1∘φb2a→=φb1φb2a→=φb1ab2→=φb1ab2→=ab2b1→=ab2b1→=φb2b1a→ for all b1b2∈A and a→∈A→. □

Suppose that A is a finite dimensional algebra over an algebraically closed field 𝔽 such that the regular representation A→ of A is completely decomposable. Then A is isomorphic to a direct sum of matrix algebras, i.e. A≅⨁λ∈AˆMdλ𝔽‾, for some index set Aˆ and some positive integers dλ, indexed by the elements of Aˆ.

		Proof.
	If A→ is completely decomposable, then by the centraliser theorem in the Schur's Lemma page, EndAA→ is isomorphic to a direct sum of matrix algebras. By proposition 1.1, Aop≅⨁λ∈AˆMdλ𝔽‾, for some set Aˆ and some positive integers dλ, indexed by the elements of Aˆ. The map ⨁λ∈AˆMdλ𝔽‾op⟶⨁λ∈AˆMdλ𝔽a⟼at, where at is the transpose of the matrix a, is an algebra isomorphism. So A is isomorphic to a direct sum of matrix algebras. □

If A is an algebra then the trace tr of the regular representation is the trace on A given by tra=TrAˆa,fora∈A, where Aˆa is the linear transformation of A induced by the action of a on A by left multiplication.

Let A=⊕λ∈AˆMdλ𝔽. The trace of the regular representation is nondegenerate if and only if the integers dλ are all nonzero in 𝔽‾. In characteristic p they could be zero.

		Proof.
	As A-modules, the regular representation A→≅⨁λ∈AˆAλ⊕dλ, where Aλ is the irreducible A-module consisting of column vectors of length dλ. For a∈A let Aλa be the linear transformation of Aλ induced by the action of a. Then the trace tr of the regular representation is given by tr=∑λ∈Aˆdλχλ,whereχAλ:A⟶𝔽‾a⟼TrAλa, where χAλ are the irreducible characters of A. Since the dλ are all nonzero the trace tr is nondegenerate. □

Reference

[HA] T. Halverson and A. Ram, Partition algebras, European Journal of Combinatorics 26, (2005), 869-921; arXiv:math/040131v2.

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