The recognition problem for table algebras and reality-based algebras
aa r X i v : . [ m a t h . R A ] A ug The recognition problem for table algebras and reality-basedalgebras
Allen Herman ∗ , Mikhael Muzychuk † , and Bangteng XuSubmitted: July 1, 2015; Revised: July 27, 2016 Abstract
Given a finite-dimensional noncommutative semisimple algebra A over C with involution,we show that A always has a basis B for which ( A, B ) is a reality-based algebra. For algebrasthat have a one-dimensional representation δ , we show that there always exists an RBA-basisfor which δ is a positive degree map. We characterize all RBA-bases of the 5-dimensionalnoncommutative semisimple algebra for which the algebra has a positive degree map, and giveexamples of RBA-bases of C ⊕ M n ( C ) for which the RBA has a positive degree map, for all n ≥ Key words :
Table algebras, C -algebras, Reality-based algebras. AMS Classification:
Primary: 05E30; Secondary: 20C15.
Let A be a ( d + 1)-dimensional involutive algebra over C , whose involution ∗ is a ring antiautomor-phism that restricts to complex conjugation on scalars. We say that the pair ( A, B ) is a reality-basedalgebra (or RBA) if there is a basis B = { b , b , . . . , b d } of A such that(i) the multiplicative identity of A is an element of B (we index the elements of B so that b isthe multiplicative identity of A );(ii) B ⊆ R B , in particular the structure constants λ ijk generated by the basis B in the expressions b i b j = d P k =0 λ ijk b k are all real numbers;(iii) B ∗ = B , so ∗ induces a transposition on the set { , , . . . , d } given by b i ∗ = ( b i ) ∗ for all b i ∈ B ;(iv) λ ij = 0 ⇐⇒ j = i ∗ ; and(v) λ ii ∗ = λ i ∗ i > ∗ The author acknowledges the support of an NSERC Discovery Grant. † The author acknowledges the support of the Wilson Endowment of Eastern Kentucky University emark. In earlier treatments of reality-based algebras in the literature, the involution of thedefinition is assumed to be C -linear. Since we have a ∗ -fixed basis, this is consistent here with ¯ ∗ ,the composition of our involution with complex conjugation on scalars.If B is a finite basis of an involutive algebra A satisfying these properties, we will say that B is an RBA-basis of A . If the structure constants relative to the RBA-basis B are integers (rationalnumbers), then we will say that the RBA-basis is integral (rational). We can similarly refer to theRBA-basis as being R -integral for any subring R of the real numbers.An RBA ( A, B ) has a degree map if there is an algebra homomorphism δ : A → C such that δ ( b i ) = δ ( b ∗ i ) ∈ R × for all b i ∈ B . This degree map is said to be positive if δ ( b i ) > b i ∈ B .When there is a positive degree map, it will be the unique algebra homomorphism A → C thatis positive on elements of B . An RBA-basis for an RBA with positive degree map is said to be standard when δ ( b i ) = λ ii ∗ for all i = 0 , , . . . , d .For any algebra, a C -linear map τ : A → C is called a feasible trace when it satisfies τ ( xy ) = τ ( yx ) for all x, y ∈ A . An RBA with positive degree map has a standard feasible trace , given by τ ( P i x i b i ) = δ ( B + ) x for all P i x i b i of A that are expressed in terms of the basis B = { b , b , . . . , b d } .This standard feasible trace satisfies τ ( x ∗ x ) > x ∈ A , and so it induces a nonde-generate R -bilinear form on A .For convenience we will say that the RBA-basis for an RBA with positive degree map is an RBA δ -basis . A table algebra is an RBA with a positive degree map for which the structure constantswith respect to its RBA-basis are all nonnegative. We will say that the distinguished basis of tablealgebra is a TA-basis. A commutative RBA with a degree map is a C -algebra .RBAs, C -algebras and table algebras have significant structural advantages that allow them tobehave more like groups than rings. To get an impression of this phenomenon, we direct the reader’sattention to [2], [3], [4], and [7]. It is of fundamental importance, therefore, to be able to determinewhether or not a semisimple algebra over C has an RBA-basis, and if so, to characterize its RBA, C -algebra, or table algebra structures. For commutative semisimple algebras existence of the RBA-bases is not an issue because a finite abelian group will be a basis. But for noncommutative algebrasit requires a nontrivial construction. For example, the algebra M ( C ) with the conjugate-transposeinvolution is one example of an RBA, since (cid:26)(cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) (cid:21) , (cid:20) − (cid:21)(cid:27) is an RBA-basis of M ( C ). Of course, M ( C ) has no chance to have a positive degree map becauseit has no one-dimensional algebra representation. (This observation was made by Blau in [4].)Our main results give a full account of the existence of RBA- and RBA δ - bases for finite-dimensional semisimple algebras, and information as to whether these bases can be integral or ra-tional. We start by giving examples of rational RBA-bases of M n ( C ) under the conjugate-transposeinvolution for all n >
1. By applying the circle product operation we show that any semisimplealgebra over C has a rational RBA-basis. This is not true for semisimple algebras over R in general,since the real quaternion algebra with its usual involution does not have an RBA-basis. In the fourthsection we use character theory to show that noncommutative algebras of the form C ⊕ M n ( C ) with n > δ -bases. In the fifth section we characterize all of the RBA δ -basesof the noncommutative 5-dimensional algebra C ⊕ M ( C ) with the conjugate-transpose involution,and give an example of a rational table algebra basis of this algebra. In the last section we constructan RBA δ -bases for C ⊕ M m ( C ) for every m ≥ Q ( √ m ).2 Rational RBA-bases for M n ( C ) We begin by constructing examples of rational RBA-bases of the algebra M n ( C ) with respect tothe conjugate-transpose involution. Our preference is to find RBA-bases whose structure constantslie in as small a ring as possible. An integral RBA-basis is suitable for use with any coefficient ring,and a rational RBA-basis will produce an RBA structure over any field of characteristic zero.The first lemma will reduce the problem to the commutative subalgebra consisting of diagonalmatrices. We will write E i,j for the elementary matrix whose ( i, j )-entry is 1 and all of its otherentries are 0. Lemma 1.
Suppose D is an RBA-basis of the commutative subalgebra of diagonal matrices in M n ( C ) for n ≥ , with trivial involution. Let B be the union of D with the set of all off-diagonalelementary matrices. Then B is an RBA-basis of M n ( C ) with respect to the conjugate-transposeinvolution.Proof. Since E i,j E k,ℓ = δ j,k E i,ℓ where δ j,k is the Kroenecker delta, this product is either 0 when j = k , or an off-diagonal elementary matrix in B when j = k and i = ℓ , or a non-zero diagonalelementary matrix E ii in the span of D when j = k and i = ℓ . In the latter case any diagonalelementary matrix E ii is one of the primitive idempotents of the RBA ( C D , D ). By [4, Lemma2.11], the coefficient of the identity occurring in E ii will be a positive real number. Thus E ∗ i,j = E j,i .Now let D be one of the diagonal elements of B . Since the involution is trivial on D we havethat D = P ni =1 p i E i,i with all p i real. Let E j,k be an off-diagonal elementary matrix in B . Then wehave that E j,k D = p k E j,k and DE j,k = p j E j,k . This implies that the only element B of B for whichthe coefficient of I in E j,k B or BE j,k will be nonzero is E k,j . The RBA-basis properties requiredfor diagonal elements of B are inherited directly from D .It remains to construct rational RBA-bases of the n -dimensional commutative algebra D ofdiagonal n × n matrices for all n >
1. If D = { b = 1 , b , . . . , b n − } is an RBA-basis of the n -dimensional commutative semisimple algebra C n , and { e , e , . . . , e n − } is the basis of primitiveidempotents of C D , then b i = P j p i,j e j where ( p ij ) i,j is the first eigenmatrix. The map b i P j p i,j E j,j identifies D with an RBA-basis of D . Note that the fact that the identity matrix isincluded in this basis is reflected by the fact that every entry of the first row of the first eigenmatrixis a 1.It thus suffices to construct a table algebra of an arbitrary dimension n that has a rationalcharacter table. In dimensions up to 4 there are association schemes that have rational charactertables, which we can use to produce the following RBA-bases (here Diag ( v ) is the diagonal matrixwhose diagonal is the vector v ):Dimension 2: { I , Diag (1 , − } , Dimension 3: { I , Diag (1 , − , , Diag (2 , , − } . andDimension 4: { I , Diag (1 , − , − , , Diag (1 , − , , − , Diag (1 , , − , − } . For dimensions 5 or more we give a construction of a table algebra that has a rational charactertable for the given dimension. Let n ≥ n = 2where it constructs a Klein 4-group.) Define a table algebra of dimension n + 2 with basis B = { b = 1 , b , ..., b n +1 } and structure constants b i = ( n − b + ( n − b i , for i = 1 , ..., n + 1 , and b i b j = B + − b i − b j − b , for i = j. B + for b + b + ... + b n +1 .) This table algebra is the Bose-Mesner algebra of thescheme corresponding to an affine plane of order n . So, if n is a prime power this association schemedoes exist, but for other values of n the table algebra construction is still valid.Its characters are χ , χ , ..., χ n +1 where χ i ( b j ) = − i = j and χ i ( b i ) = n − i, j ≥ Theorem 2.
For all n > , M n ( C ) with the conjugate-transpose involution has a rational RBA-basis. One can ask if the 4-dimensional quaternion algebra over R with respect to its usual involutionhas an RBA-basis. However, one of the nonidentity basis elements would have to be a non-realsymmetric element with respect to the involution, and no such element exists. In this section we will show that the circle product operation introduced by Arad and Fisman [1](see also [6]) can be used to show that any semisimple involutive algebra over C has an RBA-basis.Let ( A, B ) be a RBA with RBA-basis B = { b , b , . . . , b d } and structure constants λ ijk . Suppose δ is a linear character of A that is real-valued on B , and let e δ be the corresponding centrallyprimitive idempotent of A . Let ( A , B ) be another RBA with RBA-basis B = { c , c , . . . , c h } andstructure constants β ijk . The circle product ( A ◦ δ A , B ◦ δ B ) is defined by the following:(i) A ◦ δ A is an algebra whose basis B ◦ δ B is the disjoint union of B and B \ { c } .(ii) Considered as a product in A ◦ δ A , b i b j = P k λ ijk b k for all i, j ∈ { , , . . . , d } .(iii) Considered as a product in A ◦ δ A , c i c j = P k β ijk c k for i ∈ { , . . . , h } and j ∈ { , . . . , h }\{ i ∗ } .(iv) b i c j = c j b i = δ ( b i ) c j for i ∈ { , , . . . , d } and j ∈ { , . . . , h } .(v) c i c i ∗ = β ii ∗ e δ + P k> β ijk c k for i ∈ { , . . . , h } .It is a consequence of [6, Theorem 1.1] that the circle product of a C -algebra ( A, B , δ ) having arational-valued degree map δ with an RBA ( A , B ) becomes an RBA whose RBA-basis is B ◦ δ B .From the above definition, we can see that whenever F is a subfield of the real numbers for whichthe RBA-bases of B and B are both F -integral, then B ◦ δ B will be F -integral. That the circleproduct of RBA-bases with nonnegative structure constants will be an RBA-basis with nonnegativestructure constants also follows immediately from the definition. Furthermore, if B and B bothadmit positive degree maps δ and δ , then B ◦ δ B admits the positive degree map˜ δ ( b ) = ( δ ( b ) if b ∈ B δ ( b ) if b ∈ B , for all b ∈ B ◦ δ B . We will apply the circle product operation to construct an RBA-basis of C ⊕ M n ( C ). For our C -algebra ( A, B ) we use the 2-dimensional group algebra C [ C ], with degree map given by thetrivial character of C . For the RBA ( A , B ) we use M n ( C ) with a rational RBA-basis guaranteedby Theorem 2. 4 heorem 3. Let C C be the complex group algebra of the group C = { , x } , and let δ be the trivialcharacter of the group C . Let B = { b , b , . . . , b d } be a rational RBA-basis of M n ( C ) .Then C ◦ δ B is a rational RBA-basis of C ⊕ M n ( C ) .Proof. The definition of the structure constants for the circle product basis C ◦ δ B = C ∪ ( B \ { I } )requires the centrally primitive idempotent e δ = (1 + x ) of C C . The fact that this circle productbasis is a rational RBA-basis is a consequence of [6, Theorem 1.1]. From the definition of the circleproduct in [6], C [ C ◦ δ B ] = C (1 − e δ ) ⊕ C [( B \ { b } ) ∪ { e δ } ] , which is isomorphic as an algebra to C ⊕ M n ( C ) since e δ b = be δ = 1 b = b b , for all b ∈ B \{ b } . Corollary 4.
Every finite-dimensional semisimple algebra over C has a rational RBA-basis.Proof. Induct on the number of simple components of the semisimple finite-dimensional algebra A .If A is simple, then A ≃ M n ( C ) and so it has a rational RBA-basis as observed previously.If A is not simple, let A = M n ( C ) ⊕ A , where A is a semisimple algebra with fewer components.Then m = dim ( A ) < dim ( A ). By our inductive hypothesis, A has a rational RBA-basis, call this B . By the previous theorem M n ( C ) ⊕ C is isomorphic to the circle product C [ C ] ◦ δ M n ( C ). Let δ ′ be the real linear character of C C ◦ δ M n ( C ) ≃ C ⊕ M n ( C ) with δ ′ ( M n ( C )) = 0. Then δ ′ ( e δ ) = 0.If B is a rational RBA-basis for M n ( C ), then C [( C ◦ δ B ) ◦ δ ′ B ] ≃ M n ( C ) ⊕ A , since e δ ′ b − b e δ ′ = b and bb = b b = δ ′ ( b ) b = 0, for all b ∈ B and b ∈ B \ { } . | I rr ( A ) | = 2 We will require the following well-known facts concerning the character theory of RBAs. Thesehave appeared in various forms in the literature over the years (see for example [3], [5], or [2]), butfirst appeared in this generality in work of Higman [8] describing the character theory of semisimpleinvolutive algebras with a ∗ -closed basis. Proposition 5.
Let ( A, B ) be an RBA with respect to the involution ∗ , and suppose δ is a positivedegree map on A . Let Irr ( A ) be the set of irreducible characters of A , and for each χ ∈ Irr ( A ) , let m χ be the multiplicity of χ in the standard feasible trace τ of A , and let e χ be the centrally primitiveidempotent of A for which χ ( e χ ) = χ (1) > . Then the following hold:(i) (Positive multiplicities) For all χ ∈ Irr ( A ) , m χ > .(ii) (Idempotent character formula) For all χ ∈ Irr ( A ) , e χ = m χ δ ( B + ) X i χ ( b ∗ i ) λ ii ∗ b i .(iii) (Orthogonality relations) For all χ, ψ ∈ Irr ( A ) , χ ( e ψ ) = δ χψ χ (1) . δ is an irreducible character of A , and m δ = 1. For later use, wenote that since our involution extends complex conjugation on scalars, we have e ∗ χ = e χ , for all χ ∈ Irr ( A ). When ψ, χ ∈ Irr ( A ) with ψ = χ , the fact that ψ ( xe χ ) = 0, for all x ∈ A implies that τ ( x ∗ xe χ ) = m χ χ ( x ∗ x ), for all x ∈ A . It then follows from Proposition 5(i) that χ ( x ∗ x ) ≥ x ∈ A .The referee has remarked that the next theorem is a corollary to [4, Theorem 1]. The proofprovided here is independent of this result. Theorem 6.
Let ( A, B ) be a standard integral RBA with a positive degree map.If | Irr ( A ) | = 2 , then | B | = 2 .Proof. Let
Irr ( A ) = { δ, χ } . Let e δ and e χ be the two centrally primitive idempotents of A . Wecan assume that the distinguished basis B is a standardized basis, so we have δ ( b i ) = λ ii ∗ for i = 0 , , . . . , d . Let n = δ ( B + ) be the order of B . Since B is an integral RBA basis we have that δ i ∈ Z + . By Proposition 5, we have that e δ = 1 n X i b i , and e χ = m χ n X i χ ( b ∗ i ) δ i b i , for some positive real number m χ .Since | Irr ( A ) | = 2, e + e χ = b . From this one can show that n = 1 + m χ χ ( b ), and for b i = b , χ ( b i ) = − δ i ∗ m χ . In particular, χ ( b i ) is a negative rational number when b i = b . Since χ ( b i ) is analgebraic integer whenever the structure constants for the basis B are integers, all of the χ ( b i )’s arein fact integers, and so χ ( b i ) ≤ − i > X i χ ( b i ) = χ ( b ) + d X i =1 χ ( b i ) ≤ χ ( b ) − ( | B | − . Since | B | = 1 + χ ( b ) , it follows that χ ( b ) ≥ χ ( b ) . Since χ ( b ) is a positive integer, this forces χ ( b ) = 1, and hence | B | = 2, as required.One interpretation of the preceding result is that any noncommutative semisimple algebra thathas an integral RBA δ -basis must have at least 3 simple components. In particular, the noncommu-tative 5-dimensional semisimple algebra over C does not have an integral RBA δ -basis. -dimensional semisimple algebra The results of Section 4 do not tell us if the algebras C ⊕ M n ( C ) for n ≥ A = C ⊕ M ( C ). Lemma 7.
Let A = C ⊕ M ( C ) , and let δ be the algebra projection map onto its one-dimensionalcomponent. Suppose B is an RBA-basis of A for which δ takes positive values on B ; i.e. δ is apositive degree map. Then the algebra R B is isomorphic to R ⊕ M ( R ) and, up to a change of basis, ∗ acts on M ( R ) as matrix transposition. In particular, B has exactly three ∗ -fixed elements. roof. By rescaling we can assume B = { b = 1 , b , b , b , b } is a standardized RBA δ -basis of A .Set δ ( b i ) = δ i , and let n be the order of B , so n = 1 + δ + δ + δ + δ . Since A is non-commutative,the basis B contains at least one pair b i , b ∗ i of non-symmetric elements. Therefore the number of ∗ -fixed elements of B is either 1 or 3. In the first case the dimension of ∗ -fixed subspace of R B is 3while in the second one it is equal to 4.The algebra R B is a non-commutative semisimple algebra over the reals of dimension 5. There-fore either R B ∼ = R ⊕ H or R B ∼ = R ⊕ M ( R ). If R B ∼ = R ⊕ H , then by the Skolem-Noether theoremthe action of ∗ on H has the following form: x ∗ = h − ¯ xh for some unit quaternion h ∈ H (here ¯ x is the standard quaternion conjugation). It follows from ( x ∗ ) ∗ = x that h is a scalar quaternion.Therefore h is either scalar (i.e. an element of Z ( H ) ∼ = R ) or purely imaginary quaternion. In thefirst case x ∗ = ¯ x for all x ∈ H , implying that the dimension of ∗ -fixed subspace in R B is twowhich is impossible. In the second case, we have that h ∗ = − h . Since h is a purely imaginary unitquaternion, there exists non-zero purely imaginary unit quaternion q ∈ H such that ¯ qh = − h ¯ q .For this q we have q ∗ = − ¯ q = q . Let χ be the character of the unique irreducible H -module upto isomorphism; that is, χ (1 H ) = 4 , χ ( i ) = χ ( j ) = χ ( k ) = 0. Then τ ( x ) = δ ( x ) + m χ χ ( x ), for all x ∈ R B , where m χ = n − . We know χ ( x ∗ x ) ≥ x ∈ H , but χ ( q ∗ q ) = χ ( − ¯ qq ) <
0, acontradiction. This excludes the case of R B ∼ = R ⊕ H , so we must have that R B ≃ R ⊕ M ( R ).Let ∆ : R B → M ( R ) be the two-dimensional irreducible representation of R B given byprojection to the component M ( R ). Let χ be the character corresponding to this representa-tion. By Proposition 5, n = δ (1) + m χ χ (1), so m χ = n − . We have that ∆( x ∗ ) ⊤ is a 2-dimensional irreducible representation equivalent to ∆. Thus there exists an S ∈ GL ( R ) suchthat ∆( x ∗ ) ⊤ = S − ∆( x ) S . Equivalently, ∆( x ∗ ) = S ⊤ ∆( x ) ⊤ ( S − ) ⊤ . Substituting x ∗ intead of x we obtain that ∆( x ) = ( S ⊤ S − )∆( x )( S ⊤ S − ) − holds for each x ∈ R B . Combining this togetherwith ∆( R B ) = M ( R ) we obtain that S − S ⊤ = αI for some α ∈ R . It follows from S ⊤ = αS and( S ⊤ ) ⊤ = S that α = ±
1, i.e. S is either symmetric or antisymmetric.Assume first that S is antisymmetric, that is S = (cid:20) a − a (cid:21) . A direct check shows that in thiscase the map X SX ⊤ S − , X ∈ M ( R ) has a one-dimensional space of fixed points. So, in thiscase the dimension of the ∗ -fixed subspace of R B is two, a contradiction.Assume now that S is symmetric. Then S = P ⊤ DP for some D ∈ { I , Diag (1 , − , − I } and P ∈ GL ( R ). Replacing ∆( x ) by the equivalent representation Σ( x ) := ( P − ) ⊤ ∆( x ) P ⊤ weobtain Σ( x ∗ ) = D Σ( x ) ⊤ D − . If D = ± I , then we are done. It remains to deny the case of D = Diag (1 , − tr (∆( x )∆( x ∗ )) = χ ( xx ∗ ) ≥ x ∈ R B . Since Σ : R B → M ( R ) isan epimorphism, we conclude that tr ( XDX ⊤ D − ) ≥ X ∈ M ( R ). Now choosing X = (cid:20) (cid:21) we get a contradiction.The above lemma tells us that, since R B ≃ R ⊕ M ( R ), we can replace A by an isomorphicimage whose standardized RBA δ -basis is of the form B = { b = (1 , I ) , b = ( δ , B ) = b ∗ , b = ( δ , B ) = b ∗ , b = ( δ , B ) , b = b ∗ = ( δ , B ⊤ ) } , and all entries of the matrices B , B , and B are real. Label the entries of the matrices B , B ,and B so that B = (cid:20) a bb d (cid:21) , B = (cid:20) v ww x (cid:21) , B = (cid:20) r st u (cid:21) , and B = B ⊤ . A are e δ = (1 , ) and e χ = (0 , I ). By Proposition 5 we have(1 , ) = 1 n X i ( δ i , B i )and (0 , I ) = m χ n X i tr ( B ⊤ i )(1 , δ i B i ) . These give us the conditions P i B i = 0 and P i tr ( B ⊤ i ) δ i B i = nm χ I . Since (1 ,
0) + (0 , I ) = b , thecoefficient of b i in (0 , I ) for i > , − n = m χ tr ( B ⊤ i ) nδ i for i >
0, and hence tr ( B ⊤ i ) δ i = − n − for i >
0. So our character-theoretic identities are:1 + a + v + 2 r = 0 , d + x + 2 u = 0 ,b + w + s + t = 0 , ( a + d ) δ a + ( v + x ) δ v + r + u ) δ r = nm χ , ( a + d ) δ d + ( v + x ) δ x + r + u ) δ u = nm χ , and ( a + d ) δ = ( v + x ) δ = ( r + u ) δ = − n − . The conditions for linear independence of B and these equations imply that a = d or v = x , atleast one of b or w is nonzero, and s = t . By Lemma 7, we can apply a change of basis to diagonalizethe symmetric matrix B and assume b = 0.We are able to produce RBA δ -bases with real matrix entries that satisfy all of these conditions.The main result of this section describes all of these matrix entries in terms of the degrees of basiselements and some sign choices. Since R B ≃ R ⊕ M ( R ), this theorem characterizes all standardized RBA δ -bases of C ⊕ M ( C ) up to equivalence. Theorem 8.
Suppose (cid:8) (1 , I ) , ( δ , (cid:20) a d (cid:21) ) , ( δ , (cid:20) v ww x (cid:21) ) , ( δ , (cid:20) r st u (cid:21) , ( δ , (cid:20) r ts u (cid:21) ) (cid:9) is a standardized RBA δ -basis of C ⊕ M ( C ) with respect to the conjugate-transpose involution, allof whose matrix entries are real. Let ε , ε , ε = ± be three sign choices. Then the matrix entriessatisfy the identities a = − δ n − ε p nδ ( n − − δ ) n − , d = − δ n − − ε p nδ ( n − − δ ) n − ,v = − δ n − − ε nδ δ ( n − p nδ ( n − − δ ) , x = − δ n − ε nδ δ ( n − p nδ ( n − − δ ) ,w = ε s δ δ ( n − n − − δ ) , = − δ n − − ε nδ δ ( n − p nδ ( n − − δ ) , u = − δ n − ε nδ δ ( n − p nδ ( n − − δ ) ,s = − w ε s δ n n − , and t = − w − ε s δ n n − . Conversely, given positive real numbers n , δ , δ , and δ satisfying n = 1 + δ + δ + 2 δ andthree choices of sign for ε , ε , and ε , the above identities produce an RBA δ -basis of C ⊕ M ( C ) having real matrix entries. Before beginning the proof of this theorem, we establish some preliminaries. Let τ ( P i x i b i ) = nx , x = P i x i b i ∈ A be the standard feasible trace of A . Notice that τ ( x ) = δ ( x ) + n − χ ( x ). Wedenote by B ( x ) the 2-dimensional matrix corresponding to the character χ and by r ( x ) , s ( x ) theeigenvalues of B ( x ), for all x ∈ A . Clearly χ ( x ) = r ( x ) + s ( x ). Lemma 9.
For each x ∈ A we have x ∈ span C ( b , x, B + ) , where B + = b + ... + b .Proof. The ideal ( b − n − B + ) A is isomorphic to M ( C ). Since any matrix B ∈ M ( C ) satisfies theidentity B = tr ( B ) B − det( B ) I , we conclude that(( b − n − B + ) x ) = χ ( x )( b − n − B + ) x + 12 (( χ ( x ) − χ ( x ))( b − n − B + ) . Since B + z = δ ( z ) B + for z ∈ A , after opening the brackets and collecting coefficients we obtain theresult. (Here we used the identity det( B ) = ( tr ( B ) − tr ( B )).)As a corollary we obtain that for any x ∈ A such that b , x , and B + are linearly independent,there exist uniquely determined numbers κ ( x ) , λ ( x ) , µ ( x ) such that x = κ ( x ) b + λ ( x ) x + µ ( x )( B + − b − x ) (1)Let us take x ∈ A with τ ( x ) = 0 (that is b does not appear in x ). Then comparing the coefficientof b (=applying n − τ ) in both sides gives us κ ( x ) = n − τ ( x ) = h x, x ∗ i . Applying the degreehomomorphism we get δ ( x ) = κ ( x ) + λ ( x ) δ ( x ) + µ ( x )( n − − δ ( x )).It follows from (1) that B ( x ) = ( λ ( x ) − µ ( x )) B ( x ) + ( κ ( x ) − µ ( x )) I . Hence r ( x ) + s ( x ) = λ ( x ) − µ ( x ); r ( x ) s ( x ) = µ ( x ) − κ ( x ); (2)Also 0 = τ ( x ) = δ ( x ) + n − ( r ( x ) + s ( x )) and κ ( x ) n = τ ( x ) = δ ( x ) + n − ( r ( x ) + s ( x ) ) . Thisimplies r ( x ) + s ( x ) = − δ ( x ) n − r ( x ) + s ( x ) = 2 κ ( x ) n − δ ( x ) n − . (3)From here we conclude that r ( x ) s ( x ) = ( n + 1) δ ( x ) − κ ( x ) n ( n − n − = ⇒ µ ( x ) = ( n + 1) δ ( x ) − κ ( x )( n − n − . λ ( x ) = ( n + 1) δ ( x ) − n − δ ( x ) − κ ( x )( n − n − , and { r ( x ) , s ( x ) } = − δ ( x ) n − ± p κ ( x ) n ( n − − δ ( x ) nn − . (4)If x = P i =1 k i b i , then κ ( x ) = h x, x ∗ i implies that κ ( x ) = k δ + k δ + 2 k k δ . If x = b i , i = 1 ,
2, then κ ( b i ) = δ i , λ ( b i ) = ( n + 1) δ i − n − δ i ( n − , and µ ( b i ) = ( n + 1) δ i − δ i ( n − n − . (5)If x = b i , i = 3 ,
4, then κ ( b i ) = 0, and hence λ ( b i ) = ( n + 1) δ i − n − δ i ( n − , and µ ( b i ) = ( n + 1) δ i ( n − . (6)Now it follows from the above that if x = 0 then x = ( r ( x ) + s ( x )) x − r ( x ) s ( x ) b + µ ( x ) B + = ⇒ B ( x ) = − δ ( x ) n − B ( x ) − ( n + 1) δ ( x ) − κ ( x ) n ( n − n − I . (7)Taking into account that κ ( x ) = h x, x ∗ i we conclude that B ( x ) B ( x ) + B ( x ) B ( x ) = B ( x + x ) − B ( x ) − B ( x ) = − δ ( x ) n − B ( x ) − δ ( x ) n − B ( x ) − n + 1) δ ( x ) δ ( x ) − n ( n − h x , x ∗ i ( n − I . (8) Proof of Theorem 8:
First, we substitute our RBA-basis elements into the above to establishour identities for the matrix entries.
Step 1.
Substituting x = b into (4) (notice that κ ( b ) = δ ) we obtain that { r ( b ) , s ( b ) } = − δ n − ± p δ n ( n − − δ nn − ⇒ { a, d } = − δ n − ± p δ n ( n − − δ nn − ⇒ a = − δ n − ε √ ∆ n − d = − δ n − − ε √ ∆ n − , (9)where ∆ := δ n ( n − − δ n = n ( n − − δ ) δ and ε = ± Step 2.
Substituting x = b , x = b into (8) we obtain (cid:20) av ( a + d ) w ( a + d ) w dx (cid:21) = − δ n − (cid:20) v ww x (cid:21) − δ n − (cid:20) a d (cid:21) − n + 1) δ δ ( n − I . v and x : ( av = − δ n − v − δ n − a − ( n +1) δ δ ( n − dx = − δ n − x − δ n − d − ( n +1) δ δ ( n − Thus, substituting the values of a and d given in (9) into the above equations it is straightforwardto check that ( v = − δ n − − ε nδ δ ( n − √ ∆ x = − δ n − + ε nδ δ ( n − √ ∆ Step 3.
Substituting b for x into (7) we obtain w = − v − δ vn − − ( n + 1) δ − n ( n − δ ( n − . Substiting the value of v obtained in Step 2 yields w = 2 nδ δ ( n − n − − δ ) . So w = ε s nδ δ ( n − n − − δ ) (10)where ε = ± Step 4.
Substituting the values for a , v , d , and x obtained above into the equations 1 + a + v +2 r = 0 and 1 + d + x + 2 u = 0 gives the indicated values of r and u . Step 5.
Substituting x = b − b into (7) and taking into account that δ ( b − b ) = 0 , κ ( b − b ) = h b − b , b − b i = − δ , we obtain (cid:20) s − tt − s (cid:21) = − δ nn − I = ⇒ s − t = ε r δ nn − , where ε = ± Step 6.
Using w + s + t = 0 we find that s = − w ε s δ n n − , t = − w − ε s δ n n − . This completes the proof in one direction. The other direction can be proved in a straightforward(although tedious) manner by simply calculating structure constants for the basis. Our formulasfor these structure constants are (with ε := ε ε ε , and not including those involving b ):11 = ( n +1) δ − n − δ ( n − , λ = λ = λ = ( n +1) δ − ( n − δ ( n − ,λ = λ = ( n +1) δ δ − ( n − δ ( n − , λ = λ = ( n +1) δ δ − ( n − δ ( n − ,λ = λ = ( n +1) δ δ + ε ( n − √ nδ δ ( n − , λ = λ = ( n +1) δ δ − ε ( n − √ nδ δ ( n − ,λ = λ = λ = λ = ( n +1) δ δ − ( n − δ ( n − , λ = λ = ( n +1) δ δ δ + ε ( n − δ √ nδ δ δ ( n − ,λ = λ = ( n +1) δ δ − ( n − δ − ε ( n − √ nδ δ ( n − , λ = λ = λ = λ = ( n +1) δ δ ( n − λ = λ = ( n +1) δ δ δ − ε ( n − δ √ nδ δ δ ( n − , λ = λ = ( n +1) δ δ − ( n − δ + ε ( n − √ nδ δ ( n − ,λ = λ = λ = ( n +1) δ − ( n − δ ( n − , λ = ( n +1) δ − n − δ ( n − ,λ = λ = ( n +1) δ δ δ − ε ( n − δ √ nδ δ δ ( n − , λ = λ = λ = λ = ( n +1) δ δ − ( n − δ ( n − ,λ = λ = ( n +1) δ δ − ( n − δ + ε ( n − √ nδ δ ( n − , λ = λ = λ = λ = ( n +1) δ δ ( n − λ = λ = ( n +1) δ δ δ + ε ( n − δ √ nδ δ δ ( n − , λ = λ = ( n +1) δ δ − ( n − δ − ε ( n − √ nδ δ ( n − ,λ = λ = λ = ( n +1) δ ( n − , λ = λ = λ = ( n +1) δ ( n − ,λ = λ = λ = λ = ( n +1) δ − n − δ ( n − λ = λ = ( n +1) δ − n − δ ( n − ,λ = λ = ( n +1) δ δ − ( n − δ δ − ε ( n − δ √ nδ δ δ ( n − , λ = λ = ( n +1) δ δ − ( n − δ δ + ε ( n − δ √ nδ δ δ ( n − . (cid:3) Any combination of the three sign choices in Theorem 8 is interchangeable by a change of basis.This is because conjugation by (cid:20) (cid:21) interchanges the sign choices for ε and ε and fixes the onefor ε , and conjugating by (cid:20) − (cid:21) switches the choices for ε and ε and fixes that of ε . So anycombination of the three sign choices can be achieved through a sequence of these operations.We have used Theorem 8 to look for RBA δ -bases of C ⊕ M ( C ) that have rational and/ornonnegative structure constants. Both situations occur, and can occur simultaneously. Example 10.
Choosing n = 25, δ = δ = δ = 6 with any choice of the three signs produces a rational TA-basis for C ⊕ M ( C ). The basis elements (with positive sign choices) are: b = (1 , I ), b = (6 , " − √ − − √ ) , b = (6 , " − − √
312 5 √ √ − √ ) , b = (6 , " − − √ − √ √ − √ − √ − √ ) , and b ∗ . All structure constants for this basis lie in Z [ ], and the largest denominator that occursamong them is an 8. Example 11.
Here are the elements of another RBA δ -basis of C ⊕ M ( C ) whose entries andstructure constants are all rational: b = (1 , I ), b = (1 , I ) , b = (cid:0) , (cid:20) − (cid:21) (cid:1) , b = (cid:0) , (cid:20)
29 4949 − (cid:21) (cid:1) , b = (cid:0) , (cid:20)
29 49 − − (cid:21) (cid:1) , and b ∗ . RBA δ bases for C ⊕ M n ( C ) , n ≥ In light of Theorem 8 one is almost certain that C ⊕ M n ( C ) will have an RBA δ -basis for n >
2. Inthis section we give a general construction that applies for all n ≥ Theorem 12.
Let A be a finite-dimensional semisimple algebra whose involution ∗ extends complexconjugation on scalars. Suppose χ ∈ Irr ( A ) , and let Ae χ be the simple component of A correspond-ing to χ . Identify Ae χ with a full matrix algebra M m ( C ) where m = χ (1) . If A has an RBA-basisthat admits a positive degree map, then up to a change of basis, the restriction of ∗ to Ae χ will beequal to the conjugate transpose map on M m ( C ) .Proof. Suppose B = { b , b , . . . , b d } is an RBA-basis of A that admits a positive degree map δ . Let τ be the standard feasible trace of A . By Proposition 5 and the subsequent remarks, τ = P ψ ∈ Irr ( A ) m ψ ψ with m δ = 1 all m ψ >
0, and ψ ( xx ∗ ) ≥
0, for all ψ ∈ Irr ( A ). Consider therestriction of the involution ∗ to Ae χ , which we identify with M m ( C ). Since the projection of A into this simple component is surjective, the above implies that tr ( XX ∗ ) ≥ X ∈ M m ( C ).Since ∗ extends complex conjugation on scalars, the map X ( X ∗ ) ⊤ is an algebra automorphismof M m ( C ). Therefore, X ∗ = S − X ⊤ S , for some S ∈ GL m ( C ). Replacing X by X ∗ we obtain X = ( S − S ⊤ ) X ( S − S ⊤ ) − , for all X ∈ M m ( C ), and so it follows that S − S ⊤ = αI n , for somenonzero α ∈ C .From the equation S ⊤ = αS we have that S commutes with its conjugate transpose, and so S ∗ = S − S ⊤ S = S ⊤ . Furthermore, S = ( S ⊤ ) ⊤ = ¯ ααS , so α = e iθ for some θ ∈ R . If we set H = e iθ S , then H ⊤ = H = H ∗ , and X ∗ = H − X ⊤ H for all X ∈ M m ( C ).We need to find an invertible matrix P ∈ GL m ( C ) such that P X ∗ P − = ( P XP − ) ⊤ for all X ∈ M n ( C ). It is enough to find an invertible matrix P for which P ⊤ P = H .Since H is a non-degenerate Hermitian matrix, H = U − DU for some unitary matrix U ( U − = U ⊤ ) and diagonal matrix D with nonzero real diagonal entries λ , ..., λ n . For each 1 ≤ s, t ≤ n , let Y = U − E st U , where E st is the matrix unit with ( s, t )-entry 1 and 0 entries elsewhere. Then Y ∗ Y = H − Y ⊤ HY = U − D − U Y ⊤ U − DU Y = U − D − E ts DE st U. Hence, 0 ≤ χ ( Y ∗ Y ) = tr ( D − E ts DE st ) = λ − t λ s . Therefore, all of the eigenvalues of H have the same sign. Replacing H , if necessary, by − H wemay assume that λ i > i = 1 , ..., n . Let √ D be the diagonal matrix with diagonal entries √ λ i , i = 1 , . . . , n . Let P = √ DU . Then H = P ⊤ P , as required.We will finish with a construction of an RBA δ -basis of C ⊕ M m ( C ) with respect to the involutionthat restricts to the conjugate transpose in the second component. As in the last section, it sufficesto find an RBA δ -basis of A := R ⊕ M m ( R ), which is what we will do. In what follows we write( X, Y ) for tr ( XY ⊤ ). Notice that ( ⋆, ⋆ ) is a standard Euclidean form on the vector space M m ( R ).Notice that ( XY, Z ) = (
Y, X ⊤ Z ) for all X, Y, Z ∈ M m ( R ).13 roposition 13. Let δ , ..., δ m be positive real numbers. Set n := 1 + P m i =1 δ i . Assume that thereexist m matrices B , ..., B m ∈ M m ( R ) such that(a) ( B i , B j ) = ( δ i ( n − δ i ) mn − , i = j − δ i δ j mn − , i = j (b) there exists an involutive permutation i i ′ , i = 1 , ..., m with exactly m fixed points suchthat B ⊤ i = B i ′ and δ i = δ i ′ hold for all i ;(c) P m i =1 B i = − I .Then the vectors b := (1 , I ) , b i := ( δ i , B i ) form an RBA δ -basis of ˜ A = R ⊕ M m ( R ) .Proof. Define a bilinear form h ⋆, ⋆ i on ˜ A as follows h ( x, X ) , ( y, Y ) i := n − ( xy + n − m ( X, Y )) . Also define an anti-automorphism ∗ of ˜ A by ( x, X ) ∗ = ( x, X ⊤ ). A direct check shows that(i) b ∗ i = b i ′ ;(ii) h b i , b j i = δ ij δ i (here δ ij is the Kronecker’s delta);(iii) h b i b j , b k i = h b j , b i ′ b k i Since b , b , ..., b m is a basis of the real algebra ˜ A , we obtain that b i b j = P k λ ijk b k . where λ ijk arereal numbers. Since b i ’s form an orthogonal basis of ˜ A , we get λ ijk δ k = h b i b j , b k i .Since b is the identity of A and δ = 1, we can write δ ij δ i = h b i , b j i = h b , b i ′ b j i = λ i ′ j . Thus λ ab = 0 ⇐⇒ a ′ = b and in the latter case λ aa ′ = δ a > b , ..., b m satisfies all the axioms of RBA δ basis. Proposition 14.
Assume that there exist matrices B i , i = 1 , ..., m which satisfy conditions (a)-(b)of Prop 13. Denote B := P m i =1 B i . Then the matrices ˜ B i := B i − B + I, B i )( B + I, B + I ) ( B + I ) satisfy the conditions (a)-(c) of Prop 13.Proof. Notice that the linear map L : X ˜ X := X − ( B + I,X )( B + I,B + I ) ( B + I ) is a reflection with respectto the form ( ⋆, ⋆ ). Therefore ( L ( X ) , L ( Y )) = ( X, Y ), implying ( ˜ B i , ˜ B j ) = ( B i , B j ) hereby proving(a).It follows from part (a) of Prop 13 that ( B, B ) = m . Together with ( I, I ) = m this implies that˜ B = − I . Thus P i ˜ B i = ˜ B = − I . Thus ˜ B i satisfy the properties (a) and (c). The property (b)follows from L ( X ⊤ ) = L ( X ) ⊤ (notice that B + I is a symmetric matrix).14ow we’d like to build a basis B ij of M m ( R ) which satisfies conditions (a)-(b) of Prop 14. Tosimplify calculations we take all δ i to be the same, i.e. δ k = δ >
0. In this case n = 1 + m δ andthe conditions in part (a) of Proposition 14 read as follows:( B ij , B kℓ ) = (cid:26) n − δm , ( i, j ) = ( k, ℓ ) − δm , ( i, j ) = ( k, ℓ ) (11)To build a basis which satisfies (a)-(b) we look for the matrices of the form B ij = xE ij + yJ where E ij are elementary matrices, J the m × m matrix of all 1’s, and x, y are real parameterswhich will be found later. Clearly the matrices B ij satisfy part (b) of Proposition 14. To satisfy (a)we first compute the inner products ( B ij , B kℓ ). We obtain that ( B ij , B ij ) = x + 2 xy + y m and( B ij , B kℓ ) = 2 xy + y m for ( i, j ) = ( k, ℓ ). Now (11) yields the following equations for x and yx + 2 xy + y m = n − δm , xy + y m = − δm Substracting the second equation from the first we obtain x = ± p nm . Then from the second onewe get y = m (cid:16) − x ± √ m (cid:17) . Thus there exist matrices which satisfy conditions (a)-(b). ApplyingProp 14 we conclude that A has an RBA δ -basis. This proves the main result of this section. Theorem 15.
The noncommutative algebra C ⊕ M m ( C ) with the conjugate transpose involutionhas an RBA δ -basis for all m ≥ . Corollary 16.
Every finite-dimensional semisimple algebra that has a one-dimensional simple com-ponent can be equipped with an RBA δ -basis.Proof. Suppose A ≃ C ⊕ A . If A is simple then it follows from Theorem 15 that A has an RBA δ -basis. Otherwise we can write A ≃ A ⊕ M m ( C ) where A has a one-dimensional component. Byinduction on the dimension we can assume A has an RBA δ -basis B . Let B ′ be an RBA δ -basis of C ⊕ M m ( C ). By applying properties we have for the circle product and arguing as in the proof ofCorollary 4, we find that B ◦ B ′ is an RBA δ -basis of A ◦ ( C ⊕ M m ( C )) ≃ A ⊕ M m ( C ) ≃ A. Example 17.
We will illustrate the construction for Theorem 15 in the case where m = 3 and δ = 7. In this case the construction says to take x = − √ and y = √ , and set B ij to be the 3 × √ ( − E ij + J ) for 1 ≤ i, j ≤
3. Applying the reflection mapping of Proposition 14 to each B ij produces the following list of matrices ˜ B ij , for 1 ≤ i, j ≤ B = 19 − − √ − √ − √ , ˜ B = 19 − − − √ − √ − √ − − √ − √ − √ − , ˜ B = 19 − − √ − − √ − √ − − √ − √ − √ − , ˜ B = ˜ B ⊤ B = 19 − √ − − √ − √ , ˜ B = 19 − − √ − √ − √ − − − √ − √ − √ − , ˜ B = ˜ B ⊤ , ˜ B = ˜ B ⊤ , and ˜ B = 19 − √ − √ − − √ . The set consisting of (1 , I ) and the nine (7 , ˜ B ij )’s for 1 ≤ i, j ≤ δ -basis of C ⊕ M ( C ) whose structure constants lie in the ring Z [ , √ References [1] Z. Arad and E. Fisman, On table algebras, C -algebras, and applications to finite group theory, Comm. Algebra , (1991), 2955-3009.[2] Z. Arad, E. Fisman, and M. Muzychuk, Generalized table algebras, Israel J. Math. , (1999),29-60.[3] H. Blau, Quotient structures in C -algebras, J. Algebra , (1995), 24-64; Erratum: (1995), 297-337.[4] H. Blau, Table algebras, European J. Combin. , , (2009), 1426-1455.[5] H. Blau, Association schemes, fusion rings, C -algebras, and reality-based algebras where allnontrivial multiplicities are equal, J. Algebraic Combin. , (2010), 491-499.[6] H. Blau and G. Chen, Reality-based algebras, generalized Camina-Frobenius pairs, and thenon-existence of degree maps, (2012), (4), 1547-1562.[7] H. Blau and B. Xu, Irreducible characters of wreath products in reality-based algebras andapplications to association schemes, J. Algebra , (2014), 155-172.[8] D.G. Higman, Coherent algebras, Linear Algebra Appl. , (1987), 209-239. A. Herman, Department of Mathematics and Statistics, University of Regina, Regina, SK S4A0A2, CANADA. Email:
M. Muzychuk, Department of Computer Science and Mathematics, Netanya Academic College,University St. 1, 42365, Netanya, ISRAEL. Email: [email protected]
B. Xu, Department of Mathematics and Statistics, Eastern Kentucky University, 521 LancasterAve., Richmond, KY 40475-3133, USA. Email: [email protected]@eku.edu