Representations of the Gupta-Sidki group

If p is an odd prime, then the Gupta-Sidki group gp is an infinite 2-generated p-group. It is defined in a recursive manner as a particular subgroup of the automorphism group of a regular tree of degree p. In this note, we make two observations concerning the irreducible representations of the group algebra K[gp] with K an algebraically closed field. First, when char K : p, we obtain a lower bound for the number of irreducible representations of any finite degree n. Second, when char K = p, we show that if K[gp] has one nonprincipal irreducible representation, then it has infinitely many. The proofs of these two results use similar techniques and eventually depend on the fact that the commutator subgroup Hp of gp has a normal subgroup of finite index isomorphic to the direct product of p copies of 'Hp. 1. REPRESENTATIONS OF FINITE DEGREE If p is an odd prime, then we let !p denote the Gupta-Sidki group described in [GS]. Specifically, !p is an infinite 2-generated p-group defined in a recursive manner as a subgroup of the automorphism group of a regular tree of degree p. Consequently, !p is residually finite. In fact, !p is just infinite [SI] in the sense that every nontrivial normal subgroup of !p has finite index. For our purposes, a key property of this group is that if Hp is its commutator subgroup, then 'p has a normal subgroup of finite index isomorphic to the direct product of p copies of itself (see [Si]). If G is any finitely generated group and if all finite-dimensional K-representations of G are completely reducible, then a theorem of Weil (see [F]) implies that G has only finitely many K-representations of any given finite degree n. In particular, this applies to the Gupta-Sidki group !p provided char K :& p, and motivates the following Definition. Let K be a fixed algebraically closed field and, for any group G, let fG(n) be the number of irreducible representations of K[G]of degree n. Furthermore, let FG(n) = EnIL fG(i) be the number of irreducible representations of K[G] of degree FH((n)/k and FH((n) > FG((n)/k. In particular, irm sup logn FG (n) = lrm sup logn FH (n) n--+oo n--coo (ii) fG(jn) > fH(n)/k2 for some integer j = j(n) with 1 1 and fH(n) ?& O, o, then fG(jn) > fH(n)/k2. Proof. If V is an irreducible K[H]-module of degree t, then so is V 0 g for any g E G. In this way, G/H permutes the FH(n) irreducible K[H]-modules of degree FH(n)/k. Similarly, there are at least fH(n)/k orbits of irreducible K[H]-modules of degree n, and therefore the above implies that fG(n) + fG(2n) + * + fG(kn) > fH(n)l/k. In particular, if fG(jn) is the largest of the left-hand summands, then kfG(jrn) > fH(n)7k and this is the required inequality for (ii). Note that if equality occurs here, then we must have precisely fH(n)/k orbits under the action of G/H, and therefore G/H acts semiregularly. But this implies that if V if an irreducible K[H]-module of degree n, then the induced module VG is also irreducible. Thus, we obtain at least fH(n)/k irreducible K[G]-modules of degree nk, so fH(n)/k2 = fG(jrn) > fG(kn) > fH(n)/k and hence if k > 1, then fH(n) = 0 or ox. Finally, for the reverse inequality in (i), let W be an irreducible K[G]-module of degree FG(n)/k. In particular, logn FH((n) > logn FG(n)lOgn k, so lim sup logn FH (n) > lim sup logn FG (n) n--+oo n--+oo since limn lOgn k = 0. Furthermore, the first inequality in (i) yields Flg+FG(krn) log0gFH((n)-log k logknFG(knt) +logn lg l+ognk