The Terwilliger Algebras of Group Association Schemes

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1 The Terwilliger Algebras of Group Association Schemes Eiichi Bannai Akihiro Munemasa The Terwilliger algebra of an association scheme was introduced by Paul Terwilliger [7] in order to study P-and Q-polynomial association schemes. The purpose of this paper is to discuss in detail properties of the Terwilliger algebra of the group association scheme of a finite group. We shall give bounds on the dimension of the Terwilliger algebra, and define triple regularity. Let G be a finite group, C 0 = {e}, C 1,..., C d the conjugacy classes of G. For i = 0, 1,..., d, define R i = {(x, y) yx 1 C i }. Then X (G) = (G, {R i } 0 i d ) becomes a commutative association scheme, and it is called the group association scheme of the finite group G. Define the adjacency matrix A i of the relation R i : { 1 (x, y) Ri (A i ) x,y = 0 otherwise Then there exist nonnegative integers p k ij such that A i A j = d k=0 p k ija k. This is equivalent to the relation C i C j = d k=0 p k ijc k where C i = x C i x CG and the multiplication is performed as elements of the group algebra CG. If we put A = A 0,..., A d C, then A becomes a (d+1)-dimensional subalgebra of the matrix algebra M (C), and is called the Bose Mesner algebra. It is isomorphic to the center of the group algebra. Let E 0,..., E d be the primitive idempotents of A. Since A is closed under the Hadamard multiplication, we have E i E j A, so that there exist complex numbers q k ij such that E i E j = 1 qije k k k=0 (it is known that qij k are indeed real and nonnegative, see [2]). We define the diagonal matrices Ei, A i by { 1 x (Ei Ci ) xx = 0 otherwise 1

2 Then we have and the algebra (A i ) xx = (E i ) e,x A i A j = qija k k k=0 A = A 0,..., A d C = E 0,..., E d C is called the dual Bose Mesner algebra. Now the Terwilliger algebra T is the subalgebra of M (C) generated by A and A. Since T is closed under the conjugate-transpose, T is semisimple, and one can easily see that T is non-commutative if G 1. 1 Bounds on dim T In this section we give upper and lower bounds on the dimension of the Terwilliger algebra of a group association scheme. The following identities will be used in the proof of the next lemma. See [2] for a proof. Ē i = E T i = Eî for some î {0, 1,..., d}, rank E j = (E j ) e,e, q j îk rank E j = q k ijrank E k. Lemma 1 (i) tr(e i A j E k(e l A me n) T ) = δ il δ jm δ kn p k ij C k. (ii) tr(e i A je k (E l A me n ) T ) = δ il δ jm δ kn q k ijrank E k Proof. (i) This follows directly from the definition. (ii) By the identities mentioned above, tr(e i A je k (E l A me n ) T ) = tr(e l E i A je k E n A m) = δ il δ kn tr(e i A je k A m) = δ il δ kn 2 (E i ) x,y (E j ) e,y (E k ) y,x (E m ) x,e x,y G = δ il δ kn 2 (E j (E i E T k )E m ) e,e = δ il δ kn δ jm q j iˆk (E j) e,e = δ il δ kn δ jm q k ijrank E k. This completes the proof. Consider the subspaces T 0 and T 0 defined by T 0 = span C {E i A j E k 0 i, j, k d},

3 By Lemma 1 we have T 0 = span C {E i A je k 0 i, j, k d}. dim T 0 = {(i, j, k) p k ij 0}, dim T 0 = {(i, j, k) q k ij 0}. In other words, dim T 0 is the number of triples (i, j, k) such that (C i C j ) C k. There is a one-to-one correspondence between the set of primitive idempotents {E i } 0 i d and the set of complex irreducible characters of G, say E i χ i. As shown in [2], qij k = χ i(1)χ j (1) χ k (χ (1) i χ j, χ k ) holds. Thus, dim T0 is the number of triples (i, j, k) such that (χ i χ j, χ k ) 0. Let T = End G (CG) be the centralizer algebra of the permutation representation of G acting on G itself by conjugation. The dimension of T is the number of orbits of G acting on G G by simultaneous conjugation. As is well-known, this is equal to the average of fixed points, i.e., dim T = 1 C G (a) 2 = a G i=0 C i. Theorem 2 We have the following bounds on the dimension of the Terwilliger algebra T. (i) {(i, j, k) p k ij 0} dim T. (ii) {(i, j, k) q k ij 0} dim T. (iii) dim T d i=0 / C i. Proof. These are direct consequences of T 0 T, T 0 T and T T. If G is abelian, then Theorem 2 implies dim T 0 = 2, i.e., T coincides with the full matrix algebra M (C). 2 Triple regularity If the finite group G acts transitively on the set S ijk = {(g, h) C i C j gh C k } for any i, j, k {0, 1,..., d} with S ijk, we say that G is triply transitive. Note that, since G G = i,j,k S ijk and dim T 0 = {(i, j, k) S ijk }, G is triply transitive if and only if dim T 0 = dim T. In this case T 0 = T = T holds. We call the finite group G triply regular if T 0 = T, and dually triply regular if T 0 = T. Since T 0 or T 0 generates T as an algebra, G is triply regular (resp. dually triply regular) precisely when the subspace T 0 (resp. T 0 ) is a subalgebra.

4 A combinatorial meaning of the triple regularity is as follows. i, j, k, l, m, n, the size of the set Given {z C n (y, z) R l, (x, z) R m } depends only on i, j, k, l, m, n, and is independent of the choice of (x, y) S ijk. This property, when reformulated for association schemes, plays a central role in the theory of spin models (see [4], [5]). If G is abelian, then dim T 0 = 2, so that G is triply transitive. Examples. (i) Let G = A 4 be the alternating group on four letters. G is triply regular but not triply transitive. (ii) All finite groups of order 16 are triply transitive. As for the dual triple regularity, we give a sufficient condition in terms of character products. Theorem 3 Let χ 0,..., χ d be the complex irreducible characters of the finite group G. If χ i χ j is multiplicity-free for any i, j, then T0 = T holds, in particular, G is dually triply regular. Conversely, T0 = T implies that χ i χ j is multiplicity-free for any i, j. Proof. Write χ i χ j = d k=0 N k ijχ k. Then we have (Nij) k 2 = i,j,k (χ i χ j, χ i χ j ) i,j=0 ( χ i (g)χ i (g))( χ j (g)χ j (g)) g G i=0 j=0 C G (g) 2 = 1 = 1 = Thus, if N k ij {0, 1}, then i=0 g G C i dim T = / C i i=0 = Nij k i,j,k = {(i, j, k) Nij k 0} = {(i, j, k) qij k 0} = dim T0 as desired. The converse can also be seen from the above equalities.

5 Proposition 4 Let G 1 and G 2 be finite groups. (i) If G 1 and G 2 are triply transitive, so is G 1 G 2. (ii) If G 1 and G 2 are triply regular, so is G 1 G 2. (iii) If G 1 and G 2 are dually triply regular, so is G 1 G 2. Proof. (i) This follows immediately from the definition. (ii) If G 1 and G 2 are triply regular, then T 0 (G 1 ) and T 0 (G 2 ) are subalgebras. Since T 0 (G 1 G 2 ) = T 0 (G 1 ) T 0 (G 2 ), it follows that G 1 G 2 is triply regular. (iii) Similar as (ii). Theorem 5 Let D 2n = σ, τ σ n = 1, τ 2 = 1, τστ = σ 1 be the dihedral group of order 2n. Then D 2n is triply transitive and dually triply regular. Proof. First suppose that n is odd, say n = 2m + 1. Then the conjugacy classes of D 2n are C 0 = {e}, C i = {σ i, σ i } (1 i m) and C m+1 = τ σ. Thus dim T = 2m 2 + 5m + 4. In order to compute dim T 0, let us consider the product C i C j. If 1 i m, then Also C i if j = 0 C C i C j = 0 {σ 2i, σ 2i } if j = i {σ i+j, σ i j } {σ i j, σ j i } if 1 j m and j i C m+1 if j = m + 1. C m+1 C j = { Cm+1 if 0 j m C 0 C m if j = m + 1. We can easily see that the number of triples (i, j, k) with (C i C j ) C k is (m + 2) + m(2m + 2) + (2m + 2) = 2m 2 + 5m + 4 = dim T, so that D 2n is triply transitive. To show that D 2n is dually triply regular, it suffices to prove that χ i χ j is multiplicity-free, where {χ 0,..., χ m+1 } is the set of complex irreducible characters of D 2n. We may assume χ i (1) = χ j (1) = 2, otherwise one of χ i or χ j has degree 1, so that χ i χ j is irreducible. But all irreducible characters of degree 2 are obtained by inducing a linear character of the subgroup σ to D 2n. It is now straightforward to check that χ i χ j is multiplicity-free. Hence D 2n is dually triply regular by Theorem 3. Next suppose that n is even, say n = 2m. Then the conjugacy classes of D 2n are C 0 = {e}, C i = {σ i, σ i } (1 i m 1), C m = {σ m }, C m+1 = τ σ 2, and C m+2 = τσ σ 2. Thus dim T = 2m 2 + 6m + 8. A tedious calculation similar to the case n = 2m + 1 establishes dim T 0 = dim T0 = 2m 2 + 6m + 8, hence D 2n is triply transitive and dually triply regular. We have computed dim T 0, dim T 0, and dim T for all nonabelian indecomposable finite groups of order at most 100 using GAP [6]. The results

6 are tabulated in the Appendix. We have not found any group for which dim T 0 = dim T > dim T0 holds. To conclude this section, we give a list of indecomposable finite groups of order at most 24 which are not triply transitive. Balmaceda and Oura [1] has determined the Terwilliger algebra for G = S 5 and A 5. dim T 0 dim T0 dim T dim T A SL(2, 3) S Relationship with the quantum double Let à be the complex vector space with basis G G G, and define the multiplication in à by (x, g, a)(y, h, b) = δ x 1 ga,h(xy, g, ab) and extend it linearly to Ã. Then à becomes an associative algebra. The subalgebra of à defined by D = (h, g, h) g, h G C Ã. is known as the quantum double of the finite group G (precisely speaking, the quantum double is defined for a Hopf algebra, and D is the quantum double of the group Hopf algebra of G, see [3]). On the other hand, let T = (1, g, h) g, h G C Ã, and denote by T G the G-fixed subspace of T. Then T G is isomorphic to the centralizer algebra T defined in Section 2. Therefore, à is an algebra containing both the quantum double D and the Terwilliger algebra T. References [1] J. Balmaceda and M. Oura, The Terwilliger algebras of the group association schemes of S 5 and A 5, preprint. [2] E.Bannai and T.Ito Algebraic Combinatorics I. Association schemes Benjamin/Cummings, Menlo Park, Calif., [3] V. G. Drinfel d, Quantum groups, Proc. Internat. Congr. Math. (Berkeley, Calif., 1986), Vol.1, Amer. Math. Soc., Providence, R. I., 1987, pp

7 [4] F. Jaeger, Strongly regular graphs and spin models for the Kauffman polynomial, Geom. Dedicata 44 (1992), [5] F. Jaeger, On spin models, triply regular association schemes, and duality, to appear in J. Algebraic Combinatorics. [6] M. Schönert, et.al., GAP: Groups, Algorithms and Programming, Lehrstuhl D für Mathematik, RWTH Aachen, [7] P. Terwilliger, The subconstituent algebra of an association scheme, I, II, III, J. Algebraic Combinatorics, 1 (1992), , 2 (1993), , 2 (1993),