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 = {b0, b1, b2, ..., bd} such that b0 = 1A. (A, B) is called a table algebra if the following hold. (i) The structure constants for B are nonnegative real numbers; that is, bibj = d

h=0 λijh bh with λijh ∈ R≥0, for

all bi, bj ∈ 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

bi ∈ B. (Hence i∗ is defined by bi∗ = b ∗

i .)

(iii) For all bi, bj ∈ B, λij0 = 0 if j = i∗; and λii∗0 > 0.

The degree map and order

A table algebra (A, B) has a (unique) degree map ν : A → C such that ν(bi) = ν(b ∗

i ) > 0 for all bi ∈ B.

If for any bi ∈ B, ν(bi) = λii∗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 bi ∈ B is o(bi) := ν(bi)2/λii∗0, and the order

• f any nonempty subset N of B is o(N) :=

bi∈N o(bi).

Irreducible characters

A representation of A is an algebra homomorphism Φ : A → Matn(C) such that Φ(1A) = In. A representation Φ : A → Matn(C) is called irreducible if Φ(A) acts irreducibly on Cn. 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 ζB : A → C, x → o(B)γ0, if x =

d

• i=0

γibi. Let χi, 0 ≤ i ≤ r, be the irreducible characters of A. Then (cf. Higman) ζB =

r

• i=0

miχi, mi ∈ C, where mi is called the (standard feasible) multiplicity of χi, 0 ≤ i ≤ r.

Central primitive idempotents

If χ ∈ Irr(B), and e is a central primitive idempotent of A such that χ(e) = χ(1A), then e is called the central primitive idempotent corresponding to χ. If es is the central primitive idempotent of A corresponding to χs, then es = ms

• (B)

d

• i=0

χs(b∗

i )

λii∗0 bi.

Closed subsets

For any a ∈ A with a = d

i=0 αibi, define

Supp(a) := {bi : αi = 0}. For any nonempty subsets R and L of B, define RL :=

bi∈R,bj∈L Supp(bibj), and R∗ = {b∗ i : bi ∈ 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 1A ∈ N, N∗ = N, and (CN, N) is also a table algebra, called a table subalgebra of (A, B), where CN is the C-space with basis N. if N is closed and biNb∗

i ⊆ N for any bi ∈ B, then N is called

a strongly normal closed subset of B.

Properties of multiplicities

For any χs ∈ Irr(B), let ker χs := {bi ∈ B : χs(bi) = o(bi)ns}, Z(χs) := {bi ∈ B : |χs(bi)| = o(bi)ns}. Let Oϑ(B) be the intersection of all strongly normal closed subsets of B. Oϑ(B) is called the thin residue of B. Proposition(Muzychuk, Ponomarenko, et al.) Let ns and ms be the degree and multiplicity of χs, respectively. (i) ns ≤ ms. (ii) ns = ms if and only if Oϑ(B) ⊆ ker χs. (iii) ms = 1 if and only if Z(χs) = 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(bi) | bi ∈ B}.

Wreath products

Let (A, B) and (C, D) be standard table algebras, with B = {b0 = 1A, b1, . . . , bd} and D = {d0 = 1C, d1, . . . , dt}. The tensor product (A ⊗C C, B ⊗ D) is also a standard table algebra, where B ⊗ D := {bi ⊗ dj : 0 ≤ i ≤ d, 0 ≤ j ≤ t}. Let A ≀ C be the C-space with basis B ≀ D, where B ≀ D := {b0 ⊗ dj : 0 ≤ j ≤ t} ∪ {bi ⊗ 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 = {1A}, then (A ≀ C, B ≀ D) ∼ =x (C, D).

Quotient table algebras

For any nonempty subset R of B, let R+ =

bi∈R bi.

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 := {bi//N : bi ∈ B}, bi//N := o(N)−1(NbiN)+. Then (A//N, B//N) is a standard table algebra, called the quotient table algebra of (A, B) with respect to N. For any bi ∈ B, (bi//N)∗ = b ∗

i //N, and the order

• (bi//N) = o(N)−1o(NbiN).

Theorem (Xu 2014) Let (A, B) be a standard table algebra. Then the following are equivalent. (i)

• Oϑ(B)
• = 2, and

(A, B) ∼ =x

• A//Oϑ(B), B//Oϑ(B)
• ≀
• COϑ(B), Oϑ(B)
• .

(ii) There is exactly one χs ∈ Irr(B) such that ns = ms. Furthermore, ns = 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 χs ∈ Irr(B) such that ms = 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) ≀ (CN, N).

Table algebras with fused-cneters

Let (A, B) be a table algebra. If there is a partition B0, B1, . . . , Br 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 CG the group algebra

• f G over C. Then (CG, 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 χs ∈ Irr(B) such that ns = ms. (ii) (A, B) has a fused-center, and (Z(A), Cla(B)) ∼ =x

• Z(A//Oϑ(B)), Cla(B//Oϑ(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 χs ∈ Irr(B) such that ns = ms. Then the following are equivalent. (i) |Oϑ(B)| = 2. (ii) Oϑ(B) ⊆ Cla(B) (iii) (Z(A), Cla(B)) ∼ =x

• Z(A//Oϑ(B)), Cla(B//Oϑ(B))
• ≀ (COϑ(B), Oϑ(B)).

Remark: If for any χs ∈ Irr(B), ns = ms, 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 bi ∼σ bj if χs(bi)/o(bi) = χs(bj)/o(bj) for all χs ∈ 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 cσ(N) := {S+ : S is a ∼σ-class and S ⊆ N}, and kσ(N) := |cσ(N)|. That is, kσ(N) is the number of ∼σ-classes contained in N.

Theorem (Xu 2014) Let (A, B) be a standard table algebra, and δ := |{χs ∈ Irr(B) : ns = ms}|. Then the following are equivalent. (i) (A, B) has a fused-center, and (Z(A), Cla(B)) ∼ =x

• Z(A//Oϑ(B)), Cla(B//Oϑ(B))
• ≀ (C, D)

for some commutative standard table algebra (C, D). (ii) kσ(Oϑ(B)) = δ + 1, and for any χs ∈ Irr(B) such that ns = ms, χs(bi) = 0 for any bi ∈ B \ Oϑ(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 = {b0 = 1A, b1, . . . , bd} and Irr(A) = {χ0, χ1, . . . , χr}. Then for any 0 ≤ i ≤ r and 0 ≤ j ≤ d, the (χi, bj)-entry of the character table is χi(bj). 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
• ≀ (CN, N).

Association schemes

Let X = (X, {Ri}0≤i≤d) be a d-class association scheme. (not necessarily commutative) Let A0, A1, . . . , Ad be the adjacency matrices, and A the Bose-Mesner algebra of X. (A, B) is a standard table algebra, where B := {A0, A1, . . . , Ad}. A subset T of {Ri}0≤i≤d is a closed subset of X if {Ai : Ri ∈ T} is a closed subset of B. Let T be a closed subset of X. For any x ∈ X, let xT := {y ∈ X : (x, y) ∈ Ri for some Ri ∈ T}. Then X/T := {xT : x ∈ X} forms a partition of X.

Quotient schemes and fusion schemes

For a closed subset T of X, we have the quotient scheme X//T on the set X/T. Oϑ(X) := {Ri : Ai ∈ Oϑ(B)} is a closed subset of X, and X//Oϑ(X) is a thin scheme on X/Oϑ(X). If (A, B) has a fused-center, and Cla(B) = {B+

0 , B+ 1 , . . . , B+ r }, then {Si : 0 ≤ i ≤ r} is also an

association scheme on X, where Si =

Aj∈Bi Rj, 0 ≤ i ≤ r.

(X, {Si}0≤i≤r}) is a fusion scheme of X, called the fused-center of X, and denoted by Cla(X).

Wreath products of association schemes

Let X = (X, {Ri}0≤i≤d) and Y = (Y , {Sj}0≤j≤e) be association schemes, and let A0, A1, . . . , Ad and B0, B1, . . . , Be be the adjacency matrices of X and Y, respectively. Then the adjacency matrices Cl, 0 ≤ l ≤ d + e, of the wreath product X ≀ Y are given by C0 = A0 ⊗ B0, C1 = A0 ⊗ B1, . . . , Ce = A0 ⊗ Be, Ce+1 = A1 ⊗ Jm, . . . , Ce+d = Ad ⊗ Jm, where m = |Y | and Jm is the m × m matrix whose entries are all 1. In the above ⊗ is the Kronecker product of matrices.

Remark

(i) The Bose-Mesner algebra of X ≀ Y is exactly isomorphic to the wreath product of the Bose-Mesner algebras of X and Y as table algebras. (ii) If the Bose-Mesner algebra of an association scheme Z is exactly isomorphic to the wreath product of the Bose-Mesner algebras of X and Y as table algebras, then Z is not necessarily isomorphic to X ≀ Y as association schemes. The wreath product of association schemes is stronger than the wreath product of table algebras.

Corollary Let X = (X, {Ri}0≤i≤d) be an association scheme. Then the following are equivalent. (i) There is exactly one χs ∈ Irr(X) such that ns = ms. (ii) X has a fused-center, and Cla(X) ∼ = Cla(X//Oϑ(X)) ≀ Y, where Y is a 1-class association scheme.

Corollary Let X be an association scheme. Then the following are equivalent. (i)

• Oϑ(X)
• = 2, and

X ∼ = (X//Oϑ(X)) ≀ Oϑ(X)xOϑ(X), for any x ∈ X. (ii) There is exactly one χs ∈ Irr(X) such that ns = ms. Furthermore, ns = 1.

Thank you!