Monoidal categories for the combinatorics of group representations

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1 Monoidal categories for the cobinatorics of grou reresentations Ross Street February 1998 Categories are known to be useful for organiing structural asects of atheatics. However, they are also useful in finding out what structure can be disissed (coherence theores) and hence in aiding calculations. We want to illustrate this for finite set theory, linear algebra, and grou reresentation theory. We begin with soe cobinatorial set theory. Let N denote the set of natural nubers. We identify each n Œ N with the finite set n = { j Œ N : 0 j < n }. However, we ust be careful to distinguish the cartesian roduct n = { (i, j) : 0 i <, 0 j < n } fro the isoorhic set n. Let S denote the skeletal category of finite sets; elicitly, the objects are the n Œ N and the orhiss are the functions between these sets. We need to discuss the elicit construction of finite roducts in S. Let 0 be the functions given by 0 1 are functions, and that (n, 0 : n aa and 1 : n aa n (k) = i and 1 (k) = j where k = i n + j. That 0 and, 1 ) is the categorical binary roduct of and n in S, follow fro the fact that each natural nuber 0 k < n has a unique eression in the for k = i n + j with 0 i < and 0 j < n. Moreover, each natural nuber 0 h < n has a unique eression h = i n + j + k with 0 i <, 0 j < n and 0 k <. So, referring to the diagra n n n, we see that n, 0 n n,, these last three functions by 0 n, 1 n, n, 1 0 n n, n, n n, n, n, = 0, 1 = n, n, n, n, 0 = 0 1, 1 = 1 n,, : n aa, 1 n,, obtain a categorical ternary roduct (n, 0 n,,, 1 n, 1. We denote n,, : n aa n, 2 : n aa, to n,,, 2 ) of, n, in S. The universal roerty of roduct defines a functor ƒ : S S aa S given on objects by ƒ n = n and, on functions f : aa ', g : n aa n', by (f ƒ g )(i n + j) = f (i) n' + g(j) for i Œ, j Œ n. Fro the receding rearks, it follows that the equalities ( ƒ n) ƒ = ƒ (n ƒ ), 1 ƒ = = ƒ 1 are natural in each arguent. So S becoes a strict onoidal category. There is a syetry 1

2 c, n : ƒ n aa n ƒ given by c (i, n n + j) = j + i ; it is certainly not the identity function (nor is it an involution) in general. Of course S also has finite coroducts. Suose we have a finite faily ( i ) iœr of objects i of S. For 0 i < r, we define a function i i : i aa r 1 by i i (j) = i 1 +. j Then (i i ) iœr is a faily of corojections for a categorical coroduct of the faily ( i ) iœr. For aa any faily (f i ) iœr of functions f i : i n i, we obtain a function aa f 0 +f f r 1 : n r 1 0 +n n r 1 given by (f 0 +f f r 1 )( i 1 + j) = n n i 1 + f i (j) for j Œ i. This describes a functor S r : S r aa S which is the r-fold tensor roduct of another strict onoidal structure on S. The roduct and coroduct onoidal structures on S are related by distributivity. In fact, for each Œ S, the equality ( r 1 ) ƒ = 0 ƒ r 1 ƒ is natural in all the variables 0,..., r 1 and, giving right distributivity. In articular, this ilies (by taking 0 =... = r 1 = 1) that r ƒ g = g + r : r aa r q, where g + r = g g (r ters) and g : aa q is any function. For left distributivity we have the coosite ƒ ( r 1 ) c, r -1 æææææææ Æ ( r 1 ) ƒ = 0 ƒ r 1 ƒ c c 0, r 1, ææææææææ - ææ ƒ ƒ r 1, denoted by d, 0,..., r, which is not the identity function in general. We call a category - 1 with two onoidal structures, and distributivity constraints satisfying the aios in [L] and [K], a rigoid. A rigoid is right strict when both of the onoidal structures are strict and the right distributivity constraint is an identity. The rigoids considered here will be even ore secial in that the tensor roduct usually written as a su will always be a categorical coroduct. The equalier of two functions f, g : aa n is the function h : aa where h(0) < h(1) <... < h( 1) are the eleents h(i) of with f (h(i)) = g (h(i)). We shall not discuss here coequaliers or function sets in S since they are not needed below. Now we turn to cobinatorial linear algebra over the field C of cole nubers. Write Mat for the skeletal category of atrices. That is, the objects are natural nubers, while the orhiss a : aa ', called ' -atrices, are functions a : ' aa C ; the coosite a' a : aa '' of a : aa ' and a' : ' aa '' is given by usual atri ultilication: 2

3 (a' a) i'', i =  a i, i ai, i. i Œ There is a functor G : S aa Mat defined to be the identity G (n) = n on objects while, for each function f : aa ', the atri G (f) : aa ' is such that G (f) i', i = 1 if and only if i' = f (i). The binary Kronecker roduct a ƒ b : n aa 'n' of atrices a : aa ' and b : n aan' is defined by (a ƒ b) i' n'+ j', i n+ j = a b i', i j', j. a For eale, the Kronecker roduct of the atrices Á w b : 1 aa 2 and Á y : 2 aa 2 aw a Á ay a is the atri Á : 2 aa 4. This defines a functor ƒ : Mat Mat aa Mat which is bw b Á by b given on objects by ƒ n = n. Strict associativity (a ƒ b) ƒ c = a ƒ (b ƒ c) is clear; indeed, we have the forula (a ƒ b ƒ c) i' n' ' + j' ' + k', i n + j + k = a b i', i j', j c k', k for the ternary Kronecker roduct. With this structure Mat is a strict onoidal category. The Kronecker roduct is not the categorical roduct. In fact, Mat is an additive category, so direct sus rovide both categorical roduct and categorical coroduct. The direct s u a b : + n aa ' + n' of atrices a : aa ' and b : n aa n' is defined by Ïa i, i for i <, i < Ô (a b) = b for i i i', i Ì i -, i-, Ô Ó 0 otherwise. This gives another strict onoidal structure : Mat Mat aa Mat on Mat. Proosition 1 There is a syetric right-strict rigoid structure on the category Mat ade u of the Kronecker roduct and direct su onoidal structures, and such that the functor G : S aamat is syetric strict onoidal with resect to ƒ, strictly reserves the chosen finite coroducts, and reserves the distributivity isoorhiss. Proof After checking that G(f ƒ g ) = G(f ) ƒ G(g ) and G(f g ) = G(f ) G(g ), all that reains is to show is that the syetry and distributivity isoorhiss on Mat, obtained as the values of G on the syetry and distributivity isoorhiss on S, are actually natural with resect to all orhiss of Mat, and not only with resect to those in the iage of G. W e leave this calculation as an eercise for the reader, although we can see that it ust hold for 3

4 ore concetual reasons (indeed, we have arrived at our structure by transorting the rigoid structure on Mat fro the classical one on the category Vect of finite-diensional vector saces.) To find the kernel of a atri a : n aa, we reduce it to row echelon for by Gaussian eliination, and find the coluns h(0) < h(1) <... < h( 1) which do not contain a leading 1 (these are the coluns corresonding to non-echelon, or free, variables). Let k : aa n be the atri whose i-th colun is the unique solution to the hoogeneous linear syste a = 0 with h(i) = 1 and h(j) = 0 for j π i. The joint kernel of a air of atrices a : n aa, b : n aa is defined to be the kernel a of the atri Á b : n aa +. Since Gaussian eliination allows the interchange of rows, the joint kernel of the air a, b is the sae as the joint kernel of the air b, a; in other words, joint kernel deends only on the set {a, b}, not on the order. It follows that we can define the joint kernel of any finite set of atrices with fied source. Let ( ) t : Mat o aa Mat be the functor which takes each atri a : n aa to its transose a t : aa n. Notice that (a ƒ b) t = a t ƒ b t. We can use ( ) t to calculate elicit coequaliers in Mat by taking the transose of the equalier of the transose. We can also define the ho functor at : Mat o Mat aa Mat by at (a : n aa, b : aa q ) = (b ƒ a t : aa q n ). Then we have a natural bijection Mat ( ƒ n, Mat (n, at (, )) taking the atri a : n aa aa to the atri a : n with a k+ i, j = ak, in+ j. For any categories C, X, we write X C for the functor category (that is, the objects are functors F : C X aa and the orhiss are natural transforations between these. W e shall identify a grou G with the category whose only object is * and whose orhiss g : * aa * are eleents of G; coosition is grou ultilication. A atri reresentation of a grou G is a functor r : G aa Mat ; an intertwining oerator between reresentations is eactly a natural transforation between the functors. So Mat G is the category of atri reresentations of G. It is enriched in Mat by obtaining at G (r, s) as the joint kernel of the set of atrices at (r(g), 1 s(e) ) at (1 r(e), s(g)) : at (r(*), s(*)) aa at (r(*), s(*)) for g Œ G. Proosition 2 For each finite grou G there is a finite set G and an equivalence o f categories Mat G ~ Mat G. 4

5 Proof Take G to be a set of irreducible reresentations of G, one for each isoorhis class. Let i : G aa MatG be the inclusion. Then we have the functor i ~ : Mat G aa MatG given by i ~ (r)(l) = at G (l, r). Every reresentation r of G is a direct su, in Mat G, of selected irreducibles. By Schur's Lea, at G (l, r) is the ultilicity of the irreducible l in this direct su. It follows that the functor i ~ is fully faithful and essentially surjective. It follows that Mat G obtains a syetric autonoous (= coact closed) onoidal structure transorted across the equivalence fro Mat G. This leads us to the study of syetric closed onoidal structures on categories of the for Mat L where L is a finite set (regarded as a discrete category). We wish to characterie such structures in ters of structure on the set L. This is a articular case of the roble solved by Day [D]. A set L is rigged when it is equied with a distinguished eleent 1 Œ L and, for all triles l,, n Œ L, a natural nuber l, ; n, satisfying the following conditions:  ŒL 1, l; n = l, 1 ; n = l, ;, n; Ï1 Ì Ó0  for for l = n l π n =, n; l, ; ŒL for l, ŒL, l, ; n = 0 for all but a finite nuber of n. The left-hand side of the first condition is denoted by l; n and that of the second condition is denoted by l,, n;. Let N[L] denote the free coutative onoid on L; eleents are functions f : L aa N which are ero for all but a finite nuber of values; we identify each l Œ L with the function which has value 1 at l and 0 elsewhere. Rigged structures on L are in bijection with rig ultilications on the additive onoid N[L], for which the ultilicative unit is 1 Œ L. The ultilication for the rig N[L] relates to the rigged structure via the convolution forula (f * g)(n) =  l, ; nf( l) g( ). l, The rigged set is called coutative when it satisfies l, ; n =, l; n; this is recisely the condition that the rig N[L] should be coutative. A coutative rigged set is called *- autonoous when there is a function ( )* : L aa L such that l** = l and l, ; n = l, n*; *. It is called autonoous when it is *-autonoous and l, ; n = l*, *; n*. A drou is an autonoous coutative rigged set together with the following data: non-ero cole nubers l(l ), r(l ) for all l Œ L, 5,,

6 cole invertible atrices a(l,, n ; ) of sie l,, n; for all l,, n, Œ L, cole invertible atrices c(l, ; n ) of sie l, ; n for all l,, n Œ L ; satisfying the following conditions (in which c denotes the syetry for Mat ): (unity) l(l ) a(l, 1, n ; ) = r(n ) 1 l, n; (3-cocyclicity) Á 1 n, ; ƒ a( l,, ; s) Á ƒ c l, ;, n, ; 1, ; s Á, ƒ 1 l, ; a(, n, ; s) = Á a(, n, ; ) ƒ 1 l, ; s Á 1 n ƒ l s, ; a(,, ; ) Á a( l,, n; ) ƒ 1, ; s (syetry) c(, l ; n ) c(l, ; n ) = 1 l, ; n (braiding) a(, n, l ; ) Á 1 ƒ, n ; c( l, ; ) a(l,, n ; ) = Á c( l, n; ) ƒ 1, ; a(, l, n ; ) Á c( l, ; ) ƒ 1, ; (integrity)????? is a natural nuber d(l). For each drou L, we obtain a syetric onoidal category Mat [L] whose objects are eleents of N[L], whose orhiss a : f aa f ' are failies of atrices a l : f (l) aa f ' (l), whose coosition is given ointwise in Mat, and whose tensor roduct is given on objects by the convolution ultilication of N[L] and on orhiss a : f aa f ', b : g aa g ' by (a * b)(n) = l, ; n a ƒ b. l, The unit, associativity and coutativity constraints are induced in the obvious way by l(l ), r(l ), a(l,, n ; ), c(l, ; n ). l n Proosition 3 For any drou L, the onoidal category Mat [L] is autonoous. There is a strict onoidal functor w : Mat [L] aa Mat deterined by the condition w(l) = d(l). References [D] [EK] B.J. Day, On closed categories of functors, Midwest Category Seinar Reorts IV, Lecture Notes in Math. 137 (Sringer-Verlag, Berlin 1970) S. Eilenberg and G.M. Kelly, Closed categories, Proceedings of the Conference on Categorical Algebra at La Jolla (Sringer, 1966) [K] G.M. Kelly, Coherence theores for la algebras and distributive laws, Lecture Notes in Math

7 (Sringer-Verlag, 1974) [L] M.L. Lalaa, Coherence for distributivity, Lecture Notes in Math. 281 (Sringer-Verlag, 1972) [McL] S. Mac Lane, Categories for the Working Matheatician, Graduate Tets in Math. 5 (Sringer-Verlag, 1971) Macquarie University N. S. W AUSTRALIA eail: street@ath.q.edu.au We now give a general result about onoidal structures. Suose V is a onoidal category in the sense of [EK], [McL] and suose J : A V aa is a fully faithful functor such that there is an object I' of A and an isoorhis f 0 : JI', and, for all objects A, B Œ A, there eist an object CŒA and an isoorhis JA ƒ JB. By aking a choice A ƒ B of object C and a choice of isoorhis f 2;A,B : JA ƒ J(A ƒ B) for each A, B, we obtain a unique onoidal structure on A such that I' is the unit object, A ƒ B is the tensor roduct, and J, together with the isoorhiss f 0 and the f 2;A,B, becoes a strong onoidal functor J : A aa V. If the category A is skeletal (eaning that B ilies A = B), then the objects I' and A ƒ B are uniquely deterined, and the onoidal structure on A deends only on the choice of the isoorhiss f 0 and f 2;A,B. 7