ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.1056485 J. Algebra Comb. Discrete Appl. 9(1) • 1–7 Received: 10 May 2020 Accepted: 4 November 2021 Journal of Algebra Combinatorics Discrete Structures and Applications Note on the permutation group associated to E-polynomials∗ Research Article Hirotaka Imamura, Masashi Kosuda, Manabu Oura Abstract: This is a continuation of our project which focuses on E-polynomials and the related combinatorics. A pair of groups appearing in the definition of E-polynomials yields the permutation group. In this pa- per, we determine the multi-matrix structures of the centralizer algebras of the tensor representations of this permutation group. 2010 MSC: 05E10, 05E30 Keywords: Centralizer algebra, Permutation group 1. Introduction The purpose of this note is to investigate the centralizer algebras of the tensor representation of the finite group G that arises from E-polynomials in genus 1. We also discuss the association scheme which appears as the first case. Our results can be regarded as combinatorial properties of E-polynomials. The study of Eisenstein series is of interest because of its importance in number theory. We defined and studied finite analogue of Eisenstein series from combinatorial point of view. We call it an E- polynomial in some of our previous papers [8, 10, 11]. Let g be a positive integer. Then, an action of Γg = Sp(2g,Z) on the theta series induces a finite subgroup Hg of GL(2g,C). It is known that the invariant ring of Hg is closely related to the ring of modular forms for Γg. As to this direction in higher genus, see [9, 14]. There exist some results based on the fact that the weight enumerator of a self-dual and doubly even binary code is invariant under the action of Hg [3, 4, 13, 15]. The correspondence existing among the invariant theory of the finite groups, ∗ This work was supported by JSPS KAKENHI Grant Numbers 17K05164, 19K03398. Hirotaka Imamura; PFU Limited, Japan (email: h.i.ishikawa01@gmail.com). Masashi Kosuda; Department of Engineering, Yamanashi University, Japan (email: mkosuda@yamanashi.ac.jp). Manabu Oura (Corresponding Author); Institute of Science and Engineering, Kanazawa University, Japan (email: oura@se.kanazawa-u.ac.jp). 1 https://orcid.org/0000-0003-0028-6785 https://orcid.org/0000-0002-0498-9667 H. Imamura et.al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 1–7 the theory of modular forms, and combinatorics such as coding theory motivates the notion and study of E-polynomials. Let Kg be a subgroup of Hg fixing x0. An E-polynomial of degree ` in degree g is defined by ϕ`(xa : a ∈ F g 2) = 1 |Hg| ∑ Kg\Hg3σ (σx0) `. The rings generated by E-polynomials have been studied in [10, 11] (cf. [8]). Moreover, the representation theory of H1 is investigated and the centralizer rings of the tensor representation of H1 are determined in [6]. Since the character table of H1 is used in this study, it is reproduced at the end of this paper from [6]. As usual, let C denote the complex number field. We denote by Md the matrix algebra of degree d over C. For simpliticy, let nMd denote Md ⊕···⊕Md︸ ︷︷ ︸ n . 2. Preliminaries We follow the notations in [6]. Let H1 be the group of order 96 generated by 1+i2 ( 1 −1 1 1 ) , ( 1 0 0 i ) and K1 = 〈 ( 1 0 0 i ) 〉 be its subgroup of orer 4. A set of representatives of K1\H1 is given by 1,T,T 2,T 3,T 4,T 5,T 6,T 7,TD,TD2,TD3,T 3D,T 3D2,T 3D3, T 5D,T 5D2,T 5D3,T 7D,T 7D2,T 7D3,TD2T,T 3D2T,T 5D2T,T 7D2T and we name each of the classes 1, 2, . . . , 24, for example, 24 = K1T 7D2T . Now, we set Ω = {1, 2, . . . , 24}, then the action of H1 on Ω gives a transitive permutation representation of H1. We observe that T and D correspond to t = (1, 2, 3, 4, 5, 6, 7, 8)(9, 14, 12, 17, 15, 20, 18, 11)(10, 21, 13, 22, 16, 23, 19, 24), and d = (2, 9, 10, 11)(4, 12, 13, 14)(6, 15, 16, 17)(8, 18, 19, 20)(21, 22, 23, 24), respectively. Consequently, we obtain a faithful permutation representation G = 〈t,d〉 of H1. The group G is an imprimitive permutation group of order 96 on the 24 points. Under the correspondence g 7→ (δαg,β)α,β∈Ω , we may regard G as a matrix group. 3. Results We follow the argument presented in the papers [6, 7]. Let us denote by χ the permutation character of G. It is known that χ(g) is the number of α ∈ Ω which is fixed by g. Proposition 3.1. χ = χ1 + χ5 + χ9 + χ10 + χ12 + χ13 + χ14 + χ15 + χ16. 2 H. Imamura et.al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 1–7 Proof. Let χ = m1χ1 + m2χ2 + · · · + m16χ16. Here we remind that χ1 is the identity character of G. Since the characters depend on the conjugacy classes, χ(C) denotes the value of χ at a conjugacy class C. Then, we have that( χ(C1) χ(C2) . . . χ(C16) ) = ( m1 m2 . . . m16 ) X, where X denotes the character table of G presented at the end of this paper. If we know the explicit values χ(Ci)’s on the left-hand side, we multiply X−1 on both sides from the right to get the result. Suppose g ∈ C for a conjugacy class C, then χ(C) is the number of α ∈ Ω which is fixed by g. Therefore the values of χ(Ci)’s can be given by( χ(C1) χ(C2) . . . χ(C16) ) = (24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 4, 0). This completes the proof of Proposition 3.1. Before proceeding further, we shall interpret this proposition from the perspective of Wielandt [17]. Let A be the centralizer ring of the matrix group G. Then Proposition 3.1 shows that A is commutative and of dimension 9 and a basis can be obtained as follows. Let G1 be the stabilizer of a point 1 ∈ Ω in G. Then G1 has 9 orbits {1},{3},{5},{7},{2, 9, 10, 11},{4, 12, 13, 14},{6, 15, 16, 17},{8, 18, 19, 20},{21, 22, 23, 24}. With each orbit ∆ we associate a matrix V(∆) = (v∆α,β) by v∆α,β = { 1 ∃g ∈ G such that 1g = β and αg −1 ∈ ∆, 0 otherwise. In particular, for ∆ = {1}, we have that V({1}) = I where I is the 24 × 24 identity matrix. Then the matrices V(∆)’s form a basis for A. Next we shall connect A with the theory of association schemes ([1]. cf. [16]). Under the ordering of the orbits given above, we denote the matrices V(∆) by A0 = I,A1, . . . ,A8. We observe that the matrices A0,A2,A8 are symmetric and tA1 = A3, tA4 = A7, tA5 = A6. Then the set of matrices X = {A0,A1, . . . ,A8} forms a non-symmetric commutative association scheme of class 8. The algebra A is also called the Bose-Mesner algebra of the association scheme X . With regards to the commutativity and the normality of Ai’s, X has another basis E0 = 124J,E1, . . . ,E8 of the primitive idempotents, where J is a 24 × 24 matrix with every entry 1. The transformation matrix P is defined by ( A0 A1 . . . A8 ) = ( E0 E1 . . . E8 ) P which can also be called the first eigenmatrix of X . In our case, the first eigenmatrix is  1 1 1 1 4 4 4 4 4 1 1 1 1 −2 −2 −2 −2 4 1 1 1 1 0 0 0 0 −4 1 i −1 −i 2 + 2i −2 + 2i −2 − 2i 2 − 2i 0 1 i −1 −i −1 − i 1 − i 1 + i −1 + i 0 1 −i −1 i 2 − 2i −2 − 2i −2 + 2i 2 + 2i 0 1 −i −1 i −1 + i 1 + i 1 − i −1 − i 0 1 −1 1 −1 2i −2i 2i −2i 0 1 −1 1 −1 −2i 2i −2i 2i 0   . 3 H. Imamura et.al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 1–7 It turns out that X is isomorphic to No.349 in the list [5]. Summing up, we have obtained the following theorem. Theorem 3.2. The centralizer ring of the transitive permutation group G1 on the 24 points is a com- mutative association scheme of degree 8 which is isomorphic to No.349 in [5]. We proceed to the centralizer algebra of the tensor representation of G. Let A(k) be the centralizer algebra of the k-th tensor representation of G. Notice A = A(1). We set χ⊗k = 16∑ i=1 d (k) i χi and −−→ d(k) = (d (k) 1 ,d (k) 2 , . . . ,d (k) 16 ). We are going to determine the coefficients d(k)i explicitly. By Proposition 3.1, we already knew −−→ d(1) = (1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1). We shall consider the matrix A such that  χχ1 χχ2 ... χχ16   = A   χ1 χ2 ... χ16   . The matrix A can be calculated explicitly as A = X.diagonal([χ(C1),χ(C2), . . . ,χ(C16)]).X−1 = X.diagonal([24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 4, 0, 4, 0]).X−1 =   1 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 0 0 2 0 1 1 1 1 1 1 2 2 2 2 0 0 1 1 0 2 1 1 1 1 2 2 1 1 2 2 0 1 1 0 1 1 2 1 0 1 2 1 2 1 2 2 0 1 0 1 1 1 1 2 1 0 2 1 1 2 2 2 1 0 0 1 1 1 0 1 2 1 1 2 1 2 2 2 1 0 1 0 1 1 1 0 1 2 1 2 2 1 2 2 0 1 1 1 1 2 2 2 1 1 3 2 2 2 3 3 1 0 1 1 1 2 1 1 2 2 2 3 2 2 3 3 1 1 1 0 2 1 2 1 1 2 2 2 3 2 3 3 1 1 0 1 2 1 1 2 2 1 2 2 2 3 3 3 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4   . We have thus gotten −−→ d(k) = −−−→ d(k−1)A (k ≥ 2). Then we have only to apply linear algebra. The matrix A is diagonalizable and we get1 P−1AP = diagonal([4, 4, 4, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]) 1 The matrices X, A, and P can be found at [12]. 4 H. Imamura et.al. / J. Algebra Comb. Discrete Appl. 9(1) (2022) 1–7 for some non-singular matrix P . We apply this calculation to −−→ d(k) = −−→ d(1)PAk−1P−1 = (ak,bk,bk,bk,ck,dk,dk,dk,ek,ek,fk,gk,gk,gk,hk,hk), where ak = 6 · 24k−2 + 3 · 4k−2, bk = 6 · 24k−2 − 4k−2, ck = 12 · 24k−2 + 2 · 4k−2, dk = 12 · 24k−2 − 2 · 4k−2, ek = 12 · 24k−2 + 2 · 4k−2, fk = 18 · 24k−2 − 3 · 4k−2, gk = 18 · 24k−2 + 4k−2, hk = 24 k−1. We have thus obtained the following multi-matrix structures of A(k). Theorem 3.3. We have that A(k) ∼= { 9M1 k = 1, Mak ⊕ 3Mbk ⊕Mck ⊕ 3Mdk ⊕ 2Mek ⊕Mfk ⊕ 3Mgk ⊕ 2Mhk k ≥ 2, where ak,bk, . . . ,hk are given above. Corollary 3.4. (1) A(k) is commutative if and only if k = 1. (2) dim A(k) = 3456 · 242(k−2) + 48 · 42(k−2) (k ≥ 1). The second assertion of Corollary 3.4 is obtained by taking the sum of the dimensions of the simple components. It would be interesting if we interpret this number from combinatorial point of view, see [6]. This paper is concluded with a small table of dim A(k). k 1 2 3 4 5 6 dim A(k) 9 3504 1991424 1146630144 660452081664 380420288937984 Acknowledgment: This work was supported by JSPS KAKENHI Grant Numbers 17K05164, 19K03398. The computations were done with Magma [2] and Maple. References [1] E. Bannai, T. Ito, Algebraic combinatorics I: association schemes, The Benjamin/Cummings Pub- lishing Co, California (1984). [2] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: the user language, J. Symbolic Comput. 24(3-4) (1997) 235–265. [3] M. Broué, M. Enguehard, Polynômes des poids de certains codes et fonctions thêta de certains réseaux, Ann. Sci. École Norm. Sup. 5(1) (1972) 157–181. [4] W. Duke, On codes and Siegel modular forms, Internat. Math. Res. Notices 5 (1993) 125–136. 5 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.1006/jsco.1996.0125 https://doi.org/10.24033/asens.1223 https://doi.org/10.24033/asens.1223 https://doi.org/10.1155/S1073792893000121 H. 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Discrete Appl. 9(1) (2022) 1–7 H 1 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 1 0 C 1 1 C 1 2 C 1 3 C 1 4 C 1 5 C 1 6 1 T T 2 T 3 T 4 T 6 D D T D T 2 D T 3 D T 4 D T 5 D T 6 D T 7 D 2 D 2 T 2 or d er 1 8 4 8 2 4 4 6 4 12 4 3 4 12 2 4 χ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 χ 2 1 − 1 1 − 1 1 1 − 1 1 − 1 1 − 1 1 − 1 1 1 1 χ 3 1 − i − 1 i 1 − 1 i 1 − i − 1 i 1 − i − 1 − 1 1 χ 4 1 i − 1 − i 1 − 1 − i 1 i − 1 − i 1 i − 1 − 1 1 χ 5 2 0 2 0 2 2 0 − 1 0 − 1 0 − 1 0 − 1 2 2 χ 6 2 0 − 2 0 2 − 2 0 − 1 0 1 0 − 1 0 1 − 2 2 χ 7 2 0 − 2 i 0 − 2 2 i − 1 + i 1 1 + i − i 1 − i − 1 − 1 − i i 0 0 χ 8 2 0 2 i 0 − 2 − 2 i − 1 − i 1 1 − i i 1 + i − 1 − 1 + i − i 0 0 χ 9 2 0 − 2 i 0 − 2 2 i 1 − i 1 − 1 − i − i − 1 + i − 1 1 + i i 0 0 χ 1 0 2 0 2 i 0 − 2 − 2 i 1 + i 1 − 1 + i i − 1 − i − 1 1 − i − i 0 0 χ 1 1 3 1 3 1 3 3 − 1 0 − 1 0 − 1 0 − 1 0 − 1 − 1 χ 1 2 3 − 1 3 − 1 3 3 1 0 1 0 1 0 1 0 − 1 − 1 χ 1 3 3 i − 3 − i 3 − 3 i 0 − i 0 i 0 − i 0 1 − 1 χ 1 4 3 − i − 3 i 3 − 3 − i 0 i 0 − i 0 i 0 1 − 1 χ 1 5 4 0 − 4 i 0 − 4 4 i 0 − 1 0 i 0 1 0 − i 0 0 χ 1 6 4 0 4 i 0 − 4 − 4 i 0 − 1 0 − i 0 1 0 i 0 0 7 Introduction Preliminaries Results References