Loewy Structure as a $q$-analog of Composition Factors (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics)
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(2) 136 T. Sakurai. Notation and. 2. In this. section,. we. Terminology. define q ‐composition. multiplicity,. q ‐composition. length,. and the q ‐Cartan matrix. We follow the notation and terminology of [10] unless otherwise stated. The term module refers to a finitely generated. right. module.. Definition 2.1. Let V be son. a. radical of V is denoted. of V is. rad V. over a. by defined inductively by radV ra. if n>0. module. .. .. For. right artinian ring. The integer n\geq 0 the nth. an. ,. Jacob‐ radical. =V and. \mathrm{d}^{} V= rad (\mathrm{r}\mathrm{a}\mathrm{d}^{n-1}V). We then write. \mathrm{r}\mathrm{a}\mathrm{d}_{n}V=\mathrm{r}\mathrm{a}\mathrm{d}^{n}V/\mathrm{r}\mathrm{a}\mathrm{d}^{n+1}V layer of V. (Note that these are different from the custom.) The decomposition of semisimple modules \mathrm{r}\mathrm{a}\mathrm{d}_{n}V into simple modules is referred to as the Loewy structure of V and may be visualized and call it the nth radical. as. (3.2). for instance.. Definition 2.2. Let R be sentatives of. a. right. simple R‐modules.. artinian. For. an. ring. and. S_{1}. ,. .. R‐module V and. .. S_{k} the repre‐ its composition .. ,. series. 0=V_{0}<V_{1}<\cdots<V_{t}=V (t\geq 0) we. call. c_{i}(V) :=\#\{1\leq s\leq t|V_{s}/V_{s-1}\cong S_{i}\} (1\leq i\leq k) composition multiplicity of S_{i} in V and t=c_{1}(V)+\cdots+c_{k}(V) the composition length of V The latter is denoted by \ell(V) We then call its. the. .. q ‐analog defined. .. by. c_{i}(V;q):=\displaystyle \sum_{n\geq 0}c_{i} (radn. V ) q^{n}\in \mathbb{Z}[q]. multiplicity of S_{i} in V and c_{1}(V;q)+\cdots+c_{k}(V;q) q ‐composition length of V The latter is denoted by \ell(V;q). the q ‐composition. .. .. the.
(3) 137 Loewy Structure. as a. q ‐analog of. Remark 2.3. The coefficient of that the the. S_{i}. module. simple Loewy structure. of V. q^{n}. in. (2.2). appears in the. corresponds. Composition. Factors. represents the number of times. decomposition. of. \mathrm{r}\mathrm{a}\mathrm{d}_{n}V Hence .. to the vector. $\tau$(c_{1}(V;q), \ldots, c_{k}(V;q)). .. illustration, see (3.2) and (3.3). Since c_{i}(V)=\displaystyle \lim_{q\rightarrow 1}c_{i}(V_{!}\cdot q) the Loewy structure of a module can be viewed as a q ‐analog of its composition For. ,. factors. Now it is. possible to define a q ‐analog of anything that is defined in composition multiplicity or composition length. We subsequently define and investigate a q ‐analog of the Cartan matrix. terms of. Definition 2.4. Under the notation of Definition 2.2, let P_{1} , We call projective covers of S_{1} , , S_{k} .. .. .. C_{R}:=[c_{i}(P_{j})]_{1\leq i,j\leq k}. and. [12]. .. .. ,. P_{k} be the. C_{R}(q):=[c_{i}(P_{j};q)]_{1\leq i,j\leq k}. the Cartan matrix of R and the q ‐Cartan matrix of R. Remark 2.5. Wilson. .. .. and Fuller. [6] give. respectively.. general definitions for similar concepts and call them Cartan homomorphisms for graded algebras and F‐filtered Cartan matrices respectively in the context of the Cartan determinant conjecture. Bessenrodt and Holm [4] also give essentially the same definition for the factor algebra of a path algebra by a homogeneous ideal and call it. a. more. q ‐Cartan matrix.. The q ‐analogs defined above enable us to deal with Loewy structures algebraically and to ask questions involving terms of matrix theory such as determinant.. 3 Let. Group Algebras us. show. an. example. that motivates the definition of these q ‐analogs..
(4) 138 T. Sakurai. Example bra FG. 3.1. Let G be. a. group of order 4 and consider the group. field F of characteristic 2.. The. alge‐. is not. composition length distinguish the difference between the isomorphism classes of groups of order 4; namely C_{2}\times C_{2} and C_{4} The q ‐composition length, on the other hand, is sufficient to distinguish the difference satisfactorily. over a. sufficient to. .. F 1. F. /. F. \backslash. 1. F. \backslash. F. F. F. /. 1. F. F[C_{2}\times C_{2}] F[C_{4}] \ell(F[C_{2}\times C_{2}]) \ell(F[C_{2}\times C_{2}];q) In. \ell(F[C_{4}]). =4. \ell(F[C4]; q) =1+q+q^{2}+q^{3}. =(1+q)\times(1+q). fact, these polynomials. =4. naturally obtained as the generating func‐ layers. This, of course, does not happen by chance; We restate the Jennings theorem in terms of q ‐composition length or the q ‐Cartan matrix and give some remarks. are. tions of the dimensions of radical. Theorem 3.2. finite. (Jennings [7,. p ‐group, and F. a. Theorem. field of. 3.7]).. Let p be. characteristic p. .. a. prime number, G. a. abelian p ‐group. of. Set. K_{n}:=\{g\in G|g-1\in \mathrm{r}\mathrm{a}\mathrm{d}^{n}FG\} for. an. integer n\geq. O.. Then. K_{n}/K_{n+1}. is. an. elementary. rank r_{n}\geq 0 and. \displaystyle \el (FG;q)=\prod_{n\geq 1}(\frac{1-q^{np} {1-q^{n} )^{r_{n} =\det C_{FG}(q). .. (3.1).
(5) 139 Loewy Structure. Remark 3.3. Let F be. a. as a. q ‐analog of. Composition. Factors. field of characteristic p>0 and G a finite group In modular representation theory of finite. such that p divides its order.. groups, it is well‐known that the dimension of. divisible The first. the order of the. by equality. of. (3.1). is also well‐known that. second. equality. of. Therefore it is. (3.1). Sylow. can. \det C_{FG} can. p‐subgroup. be viewed is. a. expected that. some. as a. projective FG‐module. of G. [ 9 , Theorem. is. 3.1.26].. q ‐analog. of this theorem. It. [9. 3.6.31].. power of p. be viewed. hold for q‐Cartan matrices.. as a. a. ,. q ‐analog. Lemma. The. of this theorem.. properties of Cartan. matrices also. well‐known facts about Cartan. Unfortunately, longer hold naively for q ‐Cartan matrices. For example, the group algebra FA_{5} of the alternating group A5 of degree 5 over an algebraically closed field F of characteristic 2 has the Loewy structures [5, p.52] and q ‐Cartan matrix below. matrices of group. algebras. /. are no. S_{1} S_{2} S_{3}. \backslash. 1. 1. S_{2} S_{3} S_{1} S_{1} 1. 1. 1. 1. S_{1} S_{1} S_{3} S_{2} 1. 1. 1. 1. S_{3} S_{2} S_{1} S_{1}. \backslash. /. 1. 1. S_{1} S_{2} S_{3} S_{4} P_{1} P_{2} P_{3} P_{4}. Hence. \ell(P_{21}q). p‐group. be. (3.2). C_{FA 5}(q)=\left{\begin{ar y}{l 1+2q^{}+q^{4}&q+^{3}&q+^{3}&0\ q+^{3}&q^{4}1+&q^{2}&0\ q+^{3}&q^{2}&q^{4}1+&0\ 0& 0&1 \end{ar y}\right\} and. \det C_{FA_{5}}(q). are. not divisible. by 1+q. in. \mathbb{Z}[q]. (3.3). ,. unlike the. case.. Furthermore, in general q ‐Cartan matrices of group algebras need not symmetric as Cartan matrices must [9, Theorem 2. 8.21.\mathrm{i}\mathrm{i} ], although it.
(6) 140 T. Sakurai. is the. case. defined. in. (3.3).. For. example,. the group. algebra FG of. by. G=\left\{ begin{ar y}{l 1&\mathb {F}_p\ 0&\mathb {F}_p^{\times} \end{ar y}\right\}. over a. An. 4 We. the group G. field F of characteristic p has the. matrix.. C_{FG}(q)=\left{\begin{ar y}{l q^{p-\mathrm{l}1+& q&\cdots&q^{p-2}\ q^{p-2}&1+ q^{p-1}&\cdots&q^{p-3}\ & &\vdots\ q& ^{2}&\cdots&1+ q^{p-1} \end{ar y}\right\}. appropriate generalization. Path give. following q‐Cartan. a. is. (3.4). expected.. Algebras. combinatorial. interpretation of q ‐Cartan matrices for path alge‐ us begin with a simple example to observe how. bras in Theorem 4.2. Let it should look.. Example. 4.1. Let F be. Q. =. a. field and 1. \leftarrow. Q. quiver defined by the following.. the \leftarrow. 2. .. .. .. \leftarrow. k. path algebra of Q over the field F is denoted by FQ Write S_{i} for the simple FQ ‐module that corresponds to a vertex 1\leq i\leq k The projective covers P_{i} of S_{i} have the Loewy structures described below. The. .. .. S_{k}| s_{2} 1. s_{1}. s_{1}. P_{1}. P_{2}. S_{2} .. .. .. |. S_{1} .. .. .. P_{k}.
(7) 141 Loewy Structure. Hence. FQ. has the. as a. following. q ‐analog of. Composition. Factors. Cartan matrix and q ‐Cartan matrix.. C_{FQ}=\left\{ begin{ar y}{l & 1\ 1& 1& 1 \end{ar y}\right\}C_{FQ}(q)=\left\{ begin{ar y}{l 1&q ^{k-1}q\ &1 \end{ar y}\right\}. Readers. might. see. in. Example. 4.1 that the coefficient of. q^{n} of the (i, j)-. entry of C_{FQ}(q) agrees with the number of paths from j to i of length n. This phenomenon is generalized as follows. Recall that for a finite quiver. Q the. matrix with each of its. from i to. j. is called the. adjacency. Theorem 4.2. Let F be set. \{ 1,. .. .. .. ,. k\}. and. (i, j) ‐entries equaling matrix of. field and Q adjacency matrix A.. FQ ‐modules and their. a. projective. covers. respectively. Then the q ‐Cartan pressed as the power series. case. of. finite acyclic quiver with vertex Write S_{i} and P_{i} for the simple corresponding to vertices 1\leq i\leq k a. matrix. (4.1). arrows. Q.. C_{FQ}(q)=[c_{i}(P_{j};q)]. C_{FQ}(q)=\displaystyle \sum_{n\geq 0}(^{T}A)^{n}q^{n} Remark 4.3. The q=1. the number of. can. be found in. can. be. ex‐. (4.1) [10, Corollary I.11.6]..
(8) 142 T. Sakurai. References. [1] ALPERIN, J. L. Diagrams (1980), 111‐119.. [2] BENSON,. D.. The. modules for. A_{8}. Loewy. for modules.. J. Pure. structure of the. in characteristic 2.. Appl. Algebra 16,. 2. projective indecomposable Algebra 11, 13 (1983),. Comm.. 1395‐1432.. [3] BENSON, modular. (1987),. D.. CARLSON, J. F. Diagrammatic methods representations and cohomology. Comm. Algebra 15, J.,. AND. for 1‐2. 53‐121.. [4] BESSENRODT, C.,. AND. rial invariants of derived J. Math.. 229,. 1. (2007),. HOLM, T. q ‐Cartan matrices and combinato‐ categories for skewed‐gentle algebras. Pacific. 25‐47.. [5] CARLSON,. J. F. Modules and group algebras. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter.. [6] FULLER,. K. R. The Cartan determinant and. global. dimension of Ar‐. In. rings. Azumaya algebras, actions, and modules (Bloomington, 1990?, vol. 124 of Contemp. Math. Amer. Math. Soc., Providence,. tinian. IN,. RI, 1992,. pp. 51‐72.. [7] JENNINGS,. S. A. The structure of the group ring of a p‐group modular field. Trans. Amer. Math. Soc. 50 (1941), 175‐185.. [8] KOSHITANI,. S.. On the. p‐solvable group with. Loewy p ‐length. series of the group >. 1.. algebra Comm. Algebra 13,. over a. of. a. 10. (1985),. finite. 2175‐2198.. [9] NAGAO, H., demic. TSUSHIMA, Y. Representations of finite groups. Aca‐ Press, Inc., Boston, MA, 1989. Translated from the Japanese. AND. [10] SKOWRONSKI, A.,. YAMAGATA, K. Frobenius algebras. I: Basic representation theory. Zürich: European Mathematical Society (EMS),. 2011.. AND.
(9) 143 Loewy Structure. [11] WAKI,. K.. The. as a. Loewy. q ‐analog of. Composition. structure of the. Factors. projective indecomposable. modules for the Mathieu groups in characteristic 3.. Comm.. Algebra. 21, 5 (1993), 1457‐1485.. [12] WILSON, Algebra. G. V. The Cartan map 85, 2 (1983), 390‐398.. Department of Mathematics. on. categories of graded modules. J.. and Informatics. Graduate School of Science Chiba. University 1‐33, Yayoi‐cho, Inage‐ku, Chiba‐shi, Chiba,. 263‐8522. JAPAN \mathrm{E} ‐mail address: tsakurai@math.. s.chiba‐u.ac.jp.
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