1 REPRESENTATION THEORY OF PSL (q) YAQIAO LI Following re notes from book . The im is to show the qusirndomness of PSL (q), i.e., the group hs no low dimensionl representtion. 1. Representtion Theory of Finite Groups Let G be finite group, in the following we recll some elementry representtion theory for G. Sy (π, V ) is representtion of G we men V is (finite-dimensionl) complex vector spce nd π : G GL(V ) is group homomorphism. The dimension of the representtion (π, V ) is just the dimension of V over C, we consider only finite dimensionl representtion in this note. Sy representtion (π, V ) is irreducible if V hs no nontrivil G-invrint subspce, for exmple 1-dimensionl representtions re ll irreducible. Sy the representtion is fithful if π is injective, for exmple ll nontrivil representtions for simple groups re fithful. A first result is the Schur s Lemm. Let (π, V ) nd (ρ, W ) be two representtions of G, let Hom G (π, ρ) be the vector spce of interwiners T, tht is, T : V W is liner mp nd ρ(g) T = T π(g) holds for every g G. Sy π = ρ if there exists n invertible T Hom G (π, ρ). Lemm 1.1 (Schur). Let (π, V ) nd (ρ, W ) be two finite-dimensionl nd irreducible representtions of G, then 0, π dim Hom G (π, ρ) = = ρ; 1, π = ρ. Proof. Consider the kernel nd imge of T Hom G (π, ρ), use the irreducibility. To show the dimension equls 1 when π = ρ, is equivlent to show ll T Hom G (π, ρ) re just sclrs, hence we should consider the eigenvlue λ of T, then it s equivlent to show T λi = 0 Ker(T λi) = V. Next we give some stndrd opertions of representtions, enbling us to construct new representtions from given representtions. The direct sum nd tensor product is defined in the nticipted wy s follows, (π ρ)(g)(v, w) = (π(g)v, ρ(g)w), (π ρ)(g)(v w) = π(g)v ρ(g)w, cting on spces V W nd V W, respectively. The conjugte representtion (π, V ) (recll V = Hom(V, C) is the dul spce of V ) is defined s (π (g)f)(v) = f(π(g 1 )v), f V. Note tht we cn identify the tensor spce W V with Hom(V, W ) nturlly by identifying tensor w f s liner mp from V to W s (w f)(v) = f(v)w, nd both of these two spces hve dimension dim(v ) dim(w ), hence it s esy to see the following useful fct. Lemm 1. (Equivlence of two representtions). We hve the following two representtions re equivlent (σ, Hom(V, W )) = (ρ π, W V ), where σ(g)(t ) = ρ(g)t π(g 1 ) for T Hom(V, W ). Dte: My 1, 015. McGill University, 1
2 YAQIAO LI An extremely importnt nd useful notion in representtion theory of finite groups is the chrcters. The chrcter of representtion (π, V ) is the complex vlued function χ π : G C defined s χ π (g) = tr(π(g)), the trce of π(g). Some exmples re s follows. Consider the following V G := v V : π(g)v = v, g G}, which is the subspce of V fixed by G. We show the following Lemm 1.3 (The sum of chrcter on G). We hve χ π (g) = dim V G. Proof. Consider the verge opertion A π := 1 π(g) on V, clerly it s idempotent, i.e., A π = A π. Recll tht s n idempotent mtrix we hve r(a π ) = tr(a π )(this cn be seen by tht idempotent mtrix is digonlizble nd hs eigenvlues only 0 nd 1) where r(a π ) is the rnk of A π, note r(a π ) = dim Im A π, nd it s esy to see tht Im A π = V G, hence dim V G = tr(a π ) which reduces to the desired equlity. As second exmple consider the left regulr representtion (λ G, V = CG) defined s (λ(g)f)(x) = f(g 1 x) for g, x G. In this cse it s esy to see dim V G = 1 (since V G consists only of constnt functions on G), hence we hve χ λ G (g) = dim(v G ) =. But clerly we hve χ λg (e) =, hence χ λg (g) = 0 for ll other g e. Following re some esy-to-verify fcts bout chrcters. Lemm 1.4 (Properties of chrcters). Let (π, V ) nd (ρ, W ) be two representtions of G, chrcter nd dimension χ π (e) = dim V ; we hve (by considering G-invrint inner product) if π = ρ then χ π = χ ρ ; the chrcters fter opertions, χ π (g 1 ) = χ π (g), χ π (g) = χ π (hgh 1 ) h G; χ π (g) = χ π (g 1 ) = χ π (g), χ π ρ = χ π + χ ρ, χ π ρ = χ π χ ρ. One remrk bout bove properties: the G-invrint inner product, on V cn be constructed by pplying the Weyl unitry trick on ny given inner product (, ) s follows, u, v = (π(g)u, π(g)v). Now we turn to n importnt fct. The inner product of complex vlued functions on G is defined s usul, hence we cn consider the inner product of chrcters. Theorem 1 (Orthogonlity). Let (π, V ) nd (ρ, W ) be two irreducible representtions of G, then 0, π χ π, χ ρ = = ρ; 1, π = ρ. Proof. By Schur s lemm, it s equivlent to show χ π, χ ρ = dim Hom G (π, ρ). To keep nottion consistence, we compute χ ρ, χ π insted, s follows, χ ρ, χ π = 1 χ ρ (g)χ π (g) = 1 χ ρ (g)χ π (g) = 1 χ ρ π (g) = 1 χ σ (g) = dim Hom(V, W ) G = dim Hom G (π, ρ).
3 REPRESENTATION THEORY OF PSL (q) 3 Note in the bove computtion, we hve successively used Lemm 1.4, 1., nd 1.3. The lst equlity follows by observing tht Hom(V, W ) G = Hom G (π, ρ). The orthogonlity, s usul, enbles us to decompose ny representtion (π, V ) into sum of irreducible representtions. Specificlly we hve the following structure theorem. Theorem (Decomposition of representtion into irreducibles). Let (π i, W i ), i = 1,..., k be ll the distinct irreducible representtions of G, then every representtion (π, V ) cn be decomposed uniquely s k π = m i π i, i=1 where m i = χ π, χ πi is the number of representtions equivlent to π i in the decomposition of π. In prticulr, we hve the degree formul, k = (dim W i ). i=1 Proof. The decomposition is cler. Apply it to the left regulr representtion (λ G, CG), recll previously we hve seen tht χ λg (e) = nd χ λg (g) = 0 for ll other g e, hence m i = χ λg, χ πi = 1 χ λg (g)χ πi (g) = χ πi (e) = dim W i. Evluting λ G through the decomposition formul on the identity gives the degree formul. As quick ppliction we describe the irreducible representtions of belin groups. Corollry 1 (Irreducible represettions of belin groups). If G is finite belin group, then it hs exctly distinct irreducible representtions, ll of which re of dimension one. Proof. By the degree formul, it suffices to show if (π, V ) is n irreducible representtion of G, then dim V = 1. Clerly the representtion (π, V ) is equivlent to itself (π, V ), hence by Schur Lemm 1.1, we know dim Hom G (π, π) = 1, nd ctully we hve seen tht Hom G (π, π) is set of sclrs. Now G is belin implies tht π(g) Hom G (π, π) for every g G, hence every π(g) is sclr. As result, every 1-dimensionl subspce of V is G-invrint, but (π, V ) is irreducible, hence V is itself of dimension one. Clerly in this cse (i.e., when G is belin) these distinct irreducible representtions form n orthonorml bsis for CG. For exmple when G = Z n, let ω = e πi/n, then these n distinct irreducible representtions re described by χ k : Z n C for k = 0, 1,..., n 1 where χ k (r) := ω kr for r Z n. For generl (non-belin) finite groups, it cn be shown tht the number of irreducible representtions equls to the number of conjugcy clsses of G. Now we give construction of n irreducible representtion which will be useful in the next section. A finite set X is sid to be G-spce if G cts on X, specificlly, if there is homomorphism from G to Sym(X), tht is ech element in G cn be viewed s permuttion on X. Let X be G-spce, we cn consider the permuttion representtion (λ X, CX) of G s defined by (λ X (g)f)(x) = f(g 1 x) where f CX. If we choose the chrcteristic functions of CX s bsis, then it s esy to see tht λ X (g) re just permuttion mtrices for ll g G. In prticulr we see tht χ λx : G C is ctully χ λx : G N. It s lso esy to see tht dim(cx) G equls the number of orbits of X. Consider the G-spce X X by extending the ction of G on X to X X nturlly, sy G-spce X is -trnsitive if for ny two ordered pirs (x 1, y 1 ), (x, y ) X X with x i y i for i = 1,, there exists n element g G such tht g(x 1, y 1 ) = (gx 1, gy 1 ) = (x, y ). Then X
4 4 YAQIAO LI is -trnsitive implies tht X X hs exctly two orbits: the digonl, nd the rest. Hence dim(c(x X)) G =. Lemm 1.5 (An irreducible representtion). Consider the G-invrint co-dimension one subspce of CX, W = f CX : x X f(x) = 0}, then X is -trnsitive implies λ X W is n irreducible representtion of G. Proof. If we consider the two representtions (λ X X, C(X X)) nd (λ X λ X, CX CX), it is not hrd to see tht they re ctully equivlent, hence χ λx X (g) = χ λx λ X (g) = χ λx (g). Also since W is G-invrint nd codimension one, the complement subspce V of W is just the constnt functions, hence λ X V is the trivil representtion, therefore χ λx = 1 + χ λx W. Applying Lemm 1.3 we hve dim(c(x X)) G = χ λx X (g) = χ λx (g) = (1 + χ λx W (g)). Applying the integer-vlued property of χ λx W nd dim(c(x X)) G = gives tht χ λx W, χ λx W = 1, or equivlently, λ X W is irreducible. In fct, the converse is lso true, hence X is -trnsitive is equivlent to λ X W is irreducible.. Qusirndomness of PSL (q) Given finite field K = F q of order q, recll tht the group PSL (q) is defined to be PSL (q) := SL (q)/i, I} ( ) 1 0 where I =. The order of relted groups re GL (q) = q(q 1)(q 1), SL (q) = PGL (q) = q(q 1) nd q(q 1), q is even; PSL (q) = q(q 1)/, q is odd. By elementry group theory, it cn be shown tht PSL () = S 3 the symmetric group of order 3, nd PSL (3) = A 4 the lternting group of order 4, hence both re not simple. Jordn showed in 1861 tht PSL (q) re ll simple groups for q 4. Consider now q 5 nd q is odd, then we know tht PSL (q) is simple group of order q(q 1)/. A group G is sid to be k-qusirndom if its every nontrivil unitry representtion hs dimension t lest k. Such qusirndomness re useful s demonstrted in  nd . Our im in this section is to show tht PSL (q) is qusirndom. Theorem 3 (Qusirndomness of PSL (q)). Let q 5 nd q is odd, then the group PSL (q) is (q 1)/-qusirndom, tht is, every nontrivil representtion of PSL (q) hs dimension t lest (q 1)/. For simplicity, denote K := F q to be the field. Consider subgroup H PSL (q) of order H = q(q 1)/ given s follows ( ) } b H = A = 0 1 : K, b K /I, I}. Let G GL (q) be the subgroup G = A = ( ) } b : K, b K.
5 REPRESENTATION THEORY OF PSL (q) 5 It s ( esy) to see tht H is isomorphic to subgroup ( of G ) consisting ( of ) mtrices of the form b b b. The isomorphism is given by mpping 0 1 to, using this isomorphism we view H G. Hence lterntively we cn view H s ( ) } b H = A = : K, b K. To prove theorem 3, we will determine the list of irreducible representtions of group H, nd to chieve this, we first present n irreducible representtion of G of high dimension using Lemm 1.5. Observe tht G cn be viewed s the group of ffine trnsformtions on K s follows: given x K, we hve ( ( ) ( ) b x x + b =, ) 1 1 hence we hve the ffine trnsform ction of G on K given by A : K K, x x + b, ( ) b where A = G. Therefore K cn be viewed s G-spce. Let (λ K, CK) be the permuttion representtion of G. As usul, let W = f CK : x K f(x) = 0}. be the subspce of CK of co-dimension one. Lemm.1. The representtion (λ K W, W ) is n irreducible representtion of G. Proof. By Lemm 1.5 it suffices to show K is -trnsitive. As K is field, view (x 1, y 1 ) nd (x, y ) s two points, then the mtrix which mps (x 1, y 1 ) to (x, y ) is given by the corresponding line eqution. Using this fct next we determine the irreducible representtions of H G. Lemm. (List of irreducible representtions of H). The list of irreducible representtions of H is given s follows (q 1)/ distinct representtions ρ j, j = 1,,..., (q 1)/, ll of dimension one ; two distinct representtions ρ +, ρ, both of dimension (q 1)/. Proof. First by degree formul in theorem nd using the fct tht H = q(q 1)/ we see the list is complete. We first give ll the 1-dimensionl representtions. Let K := x : x K } K be the subgroup of squres in K, we hve K = (q 1)/. Since K is n belin group, Corollry 1 sys it hs exctly (q 1)/ distinct 1-dimensionl irreducible representtions, let φ j : K C, j = 1,,..., (q 1)/ be these representtions. Observe tht there is nturl homomorphism ϕ : H K given by ϕ(a) = for A = ( b ) H, then ρ j := φ j ϕ gives the desired 1-dimensionl representtions. Second let us give the two irreducible representtions of dimension (q 1)/, which come from irreducible representtion (λ K W, W ) of G. Consider the restriction of λ K W to the subgroup H G, denote it by ρ, hence (ρ, W ) is representtion of H. For A H, recll tht the representtion ρ is defined by (ρ(a)f)(x) = f(a 1 x) where f W, x K. Since K is belin, bsis of CK is given by the q chrcters χ m : K C, m = 0, 1,..., q 1 defined by χ m (x) =
6 6 YAQIAO LI ω mx where ω = e πi/q. Clerly χ 1, χ,..., χ q } = χ m : m K } is ( bsis for W CK. Let us compute the representtion (ρ, W ) of H s follows: since A 1 = 1 ) b, we hve (ρ(a)χ m )(x) = χ m (A 1 x) = χ m ( x 1 b) = χ m ( 1 b)χ m/ (x). If m K, then m/ K, nd vice vers. Hence the two subspces W + = spnχ m : m K }, W = spnχ m : m K K }, re both H-invrint. It s then esy to verify (by clculting the inner product of chrcters) tht (ρ + = ρ W +, W + ) nd (ρ = ρ W, W ) re both irreducible. Clerly they re of dimension (q 1)/. Now we re redy to prove theorem 3. Proof. Let (π, V ) be n n-dimensionl nontrivil representtion of PSL (q), we will show n (q 1)/. Consider the restriction of π to the subgroup H PSL (q), denote it by ρ, we get representtion (ρ, V ) of group H. Applying theorem nd Lemm. we hve (q 1)/ ρ = m + ρ + m ρ m j ρ j, where m j, m +, m re the number of representtions equivlent to ρ j, ρ +, ρ, respectively. It then suffices to show m + + m 1. Since H is non-belin, its commuttor subgroup [H, H] is nontrivil. Clerly the restriction of ρ j on [H, H] re ll trivil becuse they re ll 1-dimensionl representtions (which re just homomorphisms from H to C ). Now s q 5 we know PSL (q) is simple group, hence π is fithful, hence ρ is fithful too, hence ρ(a) I n for ll I A H. In prticulr this holds for I A [H, H]. As we lredy seen ll the 1-dimensionl representtions re trivil on [H, H], hence t lest one of ρ + or ρ must pper in the decomposition of ρ, i.e., m + + m 1 s desired. References  Giulin Dvidoff, Peter Srnk, nd Alin Vlette. Elementry number theory, group theory nd Rmnujn grphs, volume 55. Cmbridge University Press, 003.  Willim T Gowers. Qusirndom groups. Combintorics, Probbility nd Computing, 17(03): , 008.  Terence To. Expnsion in groups. 54b-expnsion-in-groups/. j=1