Multiplicities of irreducible characters of table algebras and - PowerPoint PPT Presentation

Multiplicities of irreducible characters of table algebras and applications to association schemes Bangteng Xu Department of Mathematics and Statistics Eastern Kentucky University June 2 – 5, 2014 Modern Trends in Algebraic Graph Theory Villanova University

Table algebras Let A be a finite dimensional associative algebra over C , with a distinguished basis B = { b 0 , b 1 , b 2 , ..., b d } such that b 0 = 1 A . ( A , B ) is called a table algebra if the following hold. (i) The structure constants for B are nonnegative real numbers; that is, b i b j = � d h =0 λ ijh b h with λ ijh ∈ R ≥ 0 , for all b i , b j ∈ B . (ii) There is an algebra anti-automorphism (denoted by ∗ ) of A such that ( a ∗ ) ∗ = a for all a ∈ A and b ∗ i ∈ B for all b i ∈ B . (Hence i ∗ is defined by b i ∗ = b ∗ i .) (iii) For all b i , b j ∈ B , λ ij 0 = 0 if j � = i ∗ ; and λ ii ∗ 0 > 0.

The degree map and order A table algebra ( A , B ) has a (unique) degree map ν : A → C such that ν ( b i ) = ν ( b ∗ i ) > 0 for all b i ∈ B . If for any b i ∈ B , ν ( b i ) = λ ii ∗ 0 , then ( A , B ) is called a standard table algebra . Any table algebra can be rescaled to a standard table algebra. The order of any b i ∈ B is o ( b i ) := ν ( b i ) 2 /λ ii ∗ 0 , and the order of any nonempty subset N of B is o ( N ) := � b i ∈ N o ( b i ).

Irreducible characters A representation of A is an algebra homomorphism Φ : A → Mat n ( C ) such that Φ(1 A ) = I n . A representation Φ : A → Mat n ( C ) is called irreducible if Φ( A ) acts irreducibly on C n . The character afforded by Φ is the linear map χ : A → C , a �→ Tr(Φ( a )) . χ is called irreducible if Φ is irreducible. χ (1) is the degree of χ .

Feasible traces A linear map ζ : A → C is called a feasible trace if for any x , y ∈ A , ζ ( xy ) = ζ ( yx ). The standard feasible trace of ( A , B ) is d � ζ B : A → C , x �→ o ( B ) γ 0 , if x = γ i b i . i =0 Let χ i , 0 ≤ i ≤ r , be the irreducible characters of A . Then (cf. Higman) r � ζ B = m i χ i , m i ∈ C , i =0 where m i is called the ( standard feasible ) multiplicity of χ i , 0 ≤ i ≤ r .

Central primitive idempotents If χ ∈ Irr( B ), and e is a central primitive idempotent of A such that χ ( e ) = χ (1 A ), then e is called the central primitive idempotent corresponding to χ . If e s is the central primitive idempotent of A corresponding to χ s , then d χ s ( b ∗ i ) m s � e s = b i . o ( B ) λ ii ∗ 0 i =0

Closed subsets For any a ∈ A with a = � d i =0 α i b i , define Supp( a ) := { b i : α i � = 0 } . For any nonempty subsets R and L of B , define b i ∈ R , b j ∈ L Supp( b i b j ), and R ∗ = { b ∗ RL := � i : b i ∈ R } . A nonempty subset N of B is called a closed subset if N ∗ N ⊆ N . If N is a closed subset of B , then 1 A ∈ N , N ∗ = N , and ( C N , N ) is also a table algebra, called a table subalgebra of ( A , B ), where C N is the C -space with basis N . if N is closed and b i N b ∗ i ⊆ N for any b i ∈ B , then N is called a strongly normal closed subset of B .

Properties of multiplicities For any χ s ∈ Irr( B ), let ker χ s := { b i ∈ B : χ s ( b i ) = o ( b i ) n s } , Z ( χ s ) := { b i ∈ B : | χ s ( b i ) | = o ( b i ) n s } . Let O ϑ ( B ) be the intersection of all strongly normal closed subsets of B . O ϑ ( B ) is called the thin residue of B . Proposition (Muzychuk, Ponomarenko, et al.) Let n s and m s be the degree and multiplicity of χ s , respectively. (i) n s ≤ m s . (ii) n s = m s if and only if O ϑ ( B ) ⊆ ker χ s . (iii) m s = 1 if and only if Z ( χ s ) = B .

Exact isomorphisms Two table algebras ( A , B ) and ( U , V ) are called exactly isomorphic , and denoted by ( A , B ) ∼ = x ( U , V ), if there is an algebra isomorphism ϕ : A → U such that ϕ ( B ) = V , where ϕ ( B ) := { ϕ ( b i ) | b i ∈ B } .

Wreath products Let ( A , B ) and ( C , D ) be standard table algebras, with B = { b 0 = 1 A , b 1 , . . . , b d } and D = { d 0 = 1 C , d 1 , . . . , d t } . The tensor product ( A ⊗ C C , B ⊗ D ) is also a standard table algebra, where B ⊗ D := { b i ⊗ d j : 0 ≤ i ≤ d , 0 ≤ j ≤ t } . Let A ≀ C be the C -space with basis B ≀ D , where B ≀ D := { b 0 ⊗ d j : 0 ≤ j ≤ t } ∪ { b i ⊗ D + : 1 ≤ i ≤ d } . Then ( A ≀ C , B ≀ D ) is a standard table algebra, called the wreath product of ( A , B ) and ( C , D ). In particular, if B = { 1 A } , then ( A ≀ C , B ≀ D ) ∼ = x ( C , D ) .

Quotient table algebras For any nonempty subset R of B , let R + = � b i ∈ R b i . Let ( A , B ) be a standard table algebra, and N be a closed subset of B . Let A // N := C ( B // N ), the C -space with basis B // N , where B // N := { b i // N : b i ∈ B } , b i // N := o ( N ) − 1 ( N b i N ) + . Then ( A // N , B // N ) is a standard table algebra, called the quotient table algebra of ( A , B ) with respect to N . For any b i ∈ B , ( b i // N ) ∗ = b ∗ i // N , and the order o ( b i // N ) = o ( N ) − 1 o ( N b i N ).

Theorem (Xu 2014) Let ( A , B ) be a standard table algebra. Then the following are equivalent. � = 2, and � � � O ϑ ( B ) (i) ( A , B ) ∼ � A // O ϑ ( B ) , B // O ϑ ( B ) � � C O ϑ ( B ) , O ϑ ( B ) � = x ≀ . (ii) There is exactly one χ s ∈ Irr( B ) such that n s � = m s . Furthermore, n s = 1.

Corollary (Antonou, 2014) Let ( A , B ) be a commutative standard table algebra such that B is not an abelian group. Then there is exactly one χ s ∈ Irr( B ) such that m s � = 1 if and only if there is a closed subset N of B such that | N | = 2, B // N is an abelian group, and ( A , B ) ∼ = x ( A // N , B // N ) ≀ ( C N , N ).

Table algebras with fused-cneters Let ( A , B ) be a table algebra. If there is a partition B 0 , B 1 , . . . , B r of B such that Cla( B ) := { B + 0 , B + 1 , . . . , B + r } is a basis of the center Z ( A ) of A , then we say that ( A , B ) has a fused-center . If ( A , B ) has a fused-center, then ( Z ( A ) , Cla( B )) is also a table algebra, a fusion of ( A , B ). Example Let G be a finite group, and C G the group algebra of G over C . Then ( C G , G ) is a standard table algebra with a fused-center, and Cla( G ) is the set of conjugacy class sums. If the Bose-Mesner algebra of an association scheme has a fused-center, then the scheme is called a group-like scheme by Hanaki.

Theorem (Xu 2014) Let ( A , B ) be a standard table algebra. Then the following are equivalent. (i) There is exactly one χ s ∈ Irr( B ) such that n s � = m s . (ii) ( A , B ) has a fused-center, and ( Z ( A ) , Cla( B )) ∼ = x � Z ( A // O ϑ ( B )) , Cla( B // O ϑ ( B )) � ≀ ( C , D ) , where ( C , D ) is a table algebra of dimension 2.

Corollary Let ( A , B ) be a standard table algebra. Assume that there is exactly one χ s ∈ Irr( B ) such that n s � = m s . Then the following are equivalent. (i) | O ϑ ( B ) | = 2. (ii) O ϑ ( B ) ⊆ Cla( B ) (iii) ( Z ( A ) , Cla( B )) ∼ = x � Z ( A // O ϑ ( B )) , Cla( B // O ϑ ( B )) � ≀ ( C O ϑ ( B ) , O ϑ ( B )). Remark : If for any χ s ∈ Irr( B ), n s = m s , then ( A , B ) is a thin table algebra; i.e. B is a group under the multiplication of A .

The binary relation ∼ σ Define a binary relation ∼ σ on B by b i ∼ σ b j if χ s ( b i ) / o ( b i ) = χ s ( b j ) / o ( b j ) for all χ s ∈ Irr( B ) . ∼ σ is an equivalence relation. An equivalence class of ∼ σ will be simply called a ∼ σ -class . Such an equivalence relation is defined for association schemes by Hanaki. Notation : Let ( A , B ) be a table algebra, and N a normal closed subset of B . Then define c σ ( N ) := { S + : S is a ∼ σ -class and S ⊆ N } , and k σ ( N ) := | c σ ( N ) | . That is, k σ ( N ) is the number of ∼ σ -classes contained in N .

Theorem (Xu 2014) Let ( A , B ) be a standard table algebra, and δ := |{ χ s ∈ Irr( B ) : n s � = m s }| . Then the following are equivalent. (i) ( A , B ) has a fused-center, and ( Z ( A ) , Cla( B )) ∼ = x � � Z ( A // O ϑ ( B )) , Cla( B // O ϑ ( B )) ≀ ( C , D ) for some commutative standard table algebra ( C , D ). (ii) k σ ( O ϑ ( B )) = δ + 1, and for any χ s ∈ Irr( B ) such that n s � = m s , χ s ( b i ) = 0 for any b i ∈ B \ O ϑ ( B ).

Character table The character table of ( A , B ) is regarded as a matrix whose columns are indexed by the elements of B and whose rows are indexed by the irreducible characters of A . Assume that B = { b 0 = 1 A , b 1 , . . . , b d } and Irr( A ) = { χ 0 , χ 1 , . . . , χ r } . Then for any 0 ≤ i ≤ r and 0 ≤ j ≤ d , the ( χ i , b j )-entry of the character table is χ i ( b j ). If the character table of ( A , B ) has an s × t zero submatrix, then s + t ≤ | B | − 1. (Blau and Xu, 2014)

Theorem (Xu 2014) Let ( A , B ) be a standard table algebra with a fused-center. Then the following are equivalent. (i) By permuting the rows and columns if necessary, the character table of ( A , B ) has an s × t zero submatrix such that s + t = | B | − 1. (ii) There is a proper closed subset N of B such that N ⊆ Cla( B ), | N | = s + 1, and � ∼ � Z ( A ) , Cla( B ) = x � � Z ( A ) // N , Cla( B ) // N ≀ ( C N , N ) .