Word-Induced Measures on Compact Groups

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1 Word-Indued Mesures on Compt roups ene S. Kopp nd John D. Wiltshire-ordon Februry 23, 2011 rxiv: v1 [mth.r] 21 Feb 2011 Abstrt Consider group word w in n letters. For ompt group, w indues mp n nd thus pushforwrd mesure µ w on from the Hr mesure on n. We ssoite to eh word w 2-dimensionl ell omplex X(w) nd prove in Theorem 2.5 tht µ w is determined by the topology of X(w). The proof mkes use of non-belin ohomology nd Nielsen s lssifition of utomorphisms of free groups [Nie24]. Fousing on the se when X(w) is surfe, we redisover representtiontheoreti formuls for µ w tht were derived by Witten in the ontext of quntum guge theory [Wit91]. These formuls generlize result of Erdős nd Turán on the probbility tht two rndom elements of finite group ommute [ET68]. As nother orollry, we give n elementry proof tht the dimension of n irreduible omplex representtion of finite group divides the order of the group; the only ingredients re Shur s lemm, bsi ounting, nd divisibility rgument. 1 Introdution How do we mesure the degree to whih group is belin? Severl prmeters ome to mind, eh with distint flvor: (1) the index of the derived subgroup; (2) the verge size of onjugy lss; (3) the dimensions of its lrgest irreduible representtions; or (4) the probbility tht two (uniformly hosen) elements ommute. So how belin is the quternion group Q 8? Its derived subgroup hs index 4; its verge onjugy lss hs 8/5 elements; it hs irreduible representtions of dimensions 1, 1, 1, 1, nd 2. Let s hek the probbility tht two elements ommute by lulting the ommuttor of every pir of elements: 1

2 [t 1, t 2 ] 1 i j k 1 i j k i j k i j k The tble ontins the identity element extly 40 times, so the probbility tht two elements ommute is 5/8. It is no oinidene tht this number is the reiprol of the verge size of onjugy lss, sine Erdős nd Turán [ET68] prove this to be true for finite groups in generl. More generlly, we my wonder bout group identities beyond ommuttion. A group word w on n letters is n element of the free group F n. iven ny n-tuple g n of elements in some group, the universl property of F n provides n element lled w( g). If w( g) is the identity of, we sy tht w is stisfied t t. With this terminology, we my sk fundmentl question of sttistil group theory: Question 1.1. How frequently is given word stisfied in given group? Here is nottion for the quntity in question: γ (w) # {t n w(t) 1}. (2) For exmple, we hve lredy omputed γ Q8 ([g 1, g 2 ]) 40. Chnging perspetive somewht, onsider ompt group nd word w F n. We define mesure µ w on s follows: for Hr-mesurble funtion f : C, fdµ w : f(w(g 1,..., g n ))dg 1 dg n, n (3) where the integrl on the right is tken with respet to the normlized Hr mesure on n. In the finite se, µ w is relted to γ (w) by δ 1 dµ w µ w ({1}) 1 n γ (w), (4) where δ 1 denotes point-mss t the identity of. 2 Interpreting roup Words Topologilly One wy to understnd group words is topologil, sine there s nturl wy to build ell omplex out of word. If w(, b) b 1 b 1 1, for instne, we obtin the following spe. 2 (1)

3 The spe onsists of yle built out of w nd single 2-ell tthed long it, modulo the identifitions of 1-ells given by the word. The tthing mp from the boundry of the 2-ell is lled the word polygon. We strt with n esy topologil lemm bout suh spes: Lemm 2.1. Let w 1 nd w 2 be words, nd let δ 1 nd δ 2 be the word polygons. If X(w 1 ) nd X(w 2 ) re homeomorphi, then there s homotopy equivlene between the 1-skeletons for whih h δ 1 nd δ 2 re homotopi. h : Sk 1 X(w 1 ) Sk 1 X(w 2 ) (5) Proof. Let f : X(w 1 ) X(w 2 ) be the homeomorphism. Pik point x 2 X 2 tht doesn t lie in h(sk 1 X 1 ) Sk 1 X 2, nd let x 1 : f 1 (x 2 ). Punture eh spe t x i ; now f restrits to homeomorphism f between X 1 \ {x 1 } nd X 2 \ {x 2 }. Eh X i \ {x i } hs n esy deformtion retrtion to Sk i X i, given by enlrging the hole. The two 1-skeletons re now deformtion retrts of homeomorphi spes, so we get homotopy equivlene between them. The word polygon δ 1 is homotopi to smll irle round x 1, whih is tken by f to smll irle round x 2 ; this new irle is homotopi to δ 2. Thus, we hve homotopy between h δ 1 nd δ 2. For exmple, when we punture the ell omplex we exmined erlier, we see tht the blue irle is homotopi to the word polygon. b b b b 3

4 To ontinue this topologil interprettion of group words, we must find proedure nlogous to evluting word t n n-tuple g n. To this end, we lbel eh 1-ell in X(w) with the orresponding oordinte of g, obtining ellulr 1-oyle with oeffiients in. The set of suh 1-oyles inherits nturl mesure struture from the normlized Hr mesure on n. We equip the set of ohomology lsses with the nturl quotient mesure. Proposition 2.2. Let be finite onneted grph ontining n edge e 1 between distint verties. Let H be quotient spe obtined by modding out by e 1. Then, there is homotopy equivlene so tht the indued isomorphisms on non-belin ohomology re mesure-preserving. H 1 (, ) r H (6) s r s H 1 (H, ) Proof. Sy e 1 is between verties U nd V, with U V. Sy e 2,..., e m re the other edges ontining V, nd e m+1,..., e n re the remining edges. In H, nme the orresponding vertex V nd edges e 2,..., e n. Let r : H be the quotient mp. Let s be mp tking e i to e 1 e i for 2 i m nd to e i for m + 1 i n. It s esy to see tht sr nd rs re homotopy equivlent to the identity mps on nd H, respetively. (7) r (8) s (9) 4

5 Now we look t wht these mps do to oyles (with oeffiients in ). In order to write down oyles expliitly in non-belin ohomology, we must hoose diretions for the edges; for onveniene, hoose e 1 to ulminte in V, e 2,..., e m to emnte from V, nd e 2,..., e m to emnte from V. For the remining edges, put the sme (rbitrry) diretion on e i nd e i. We then hve r (g 2,..., g n ) (1, g 2,..., g n ) s (g 1,..., g n ) (g 1 g 2,..., g 1 g m, g m+1,..., g n ). (10) Sy f : H 1 (, ) C is mesurble funtion. In H 1 (, ), the oyles (g 1,..., g p ) nd (1, g1 1 g 2,..., g1 1 g m, g m+1,..., g n ) re ohomologous. We hve f(g 1,..., g p )d g f(1, g1 1 g 2,..., g1 1 g m, g m+1,..., g n )d g n n f(1, g 2,..., g n)d g dg 1 n 1 f(1, g 2,..., g n)d g n 1 f(r (g 2,..., g n))d g, (11) n 1 provided one of the integrls onverges. Similrly, f(s (g 1,..., g p ))d g f(g 1 g 2,..., g 1 g m, g m+1,..., g n )d g n n f(g 2,..., g n)d g dg 1 n 1 f(g 2,..., g n)d g. (12) n 1 Thus, r nd s re mesure-preserving. Corollry 2.3. Let be finite onneted grph. Then, there is homotopy equivlene r H (13) so tht H hs only one vertex nd the indued isomorphisms on non-belin ohomology s re mesure-preserving. H 1 (, ) r s H 1 (H, ) (14) Proof. Choose spnning tree for nd pply Proposition 2.2 repetedly until every edge in the spnning tree hs been ontrted. 5

6 Lemm 2.4. For ny homotopy equivlene of finite onneted grphs the indued isomorphism on non-belin ohomology is in ft mesure-preserving. Proof. By Lemm 2.3, we n find equivlenes 1 h 2, (15) H 1 ( 1, ) h H 1 ( 2, ) (16) 1 h 2 s H 1 r H 2 (17) so tht H 1 nd H 2 hve only one vertex. The omposition rhs is homotopy equivlene between H 1 nd H 2. It defines n isomorphism between their fundmentl groups, both of whih re the free group on the edges (nd so they both hve the sme number of edges). Choosing n rbitrry bijetion between the edge sets; this mp beomes n utomorphism of F n. All utomorphisms of F n re generted by elementry Nielsen trnsformtions on the generting set: permuting the genertors, repling genertor by its inverse, nd shering genertor by multiplying it by nother genertor [Nie24, FRS95]. The orresponding substitutions on H 1 (H 1, ) n H 1 (H 2, ) re vlid Hr substitutions. Thus, in the following digrm, (ll the mps re isomorphism nd) (rhs) s h r, r, nd s re mesure-preserving. H 1 ( 1, ) h H 1 ( 2, ) (18) s r It follows tht h is lso mesure-preserving. H 1 (H 1, ) H 1 (H 2, ) Theorem 2.5. Let w 1 nd w 2 be words giving rise to homeomorphi ell omplexes X(w i ), nd let be ompt Husdorff group. If f : C is mesurble funtion on the group, then f(w 1 ( g))d g f(w 2 ( g))d g, (19) n 1 provided one of the integrls onverges. In other words, the mesures µ w1 µ w2. Proof. By Lemm 2.1, there is homotopy equivlene between the 1-skeletons of X 1 : X(w 1 ) nd X 2 : X(w 2 ): h : Sk 1 X 1 Sk 1 X 2 (20) 6 n 2

7 under whih the word polygons orrespond. By Lemm 2.4, the indued mp h : H 1 (Sk 1 X 1, ) H 1 (Sk 1 X 2, ) (21) is mesure-preserving. The word mp w 1 : n 1 is onstnt on ohomologil equivlene lsses beuse the word polygon is yle. Therefore, f(w 1 ( g))d g f(w 1 (h ( g)))d g. (22) n 1 n 2 By the definition of the mps, f(w 1 (h ( g))) f((h w 1 )( g)), where h : Π 1 (Sk 1 X 1 ) Π 1 (Sk 2 X 1 ) (23) denotes the indued equivlene of fundmentl groupoids. Beuse the word polygons orrespond under h, we hve h w 1 w 2. Putting it ll together, f(w 1 ( g))d g f(w 2 ( g))d g. (24) n 1 n 2 3 A Representtion-Theoreti Perspetive If ρ is (ontinuous) irreduible representtion of ompt group, then the R-lgebr generted by the mtries in ρ() is simple. By the Artin-Wedderburn theorem, it is isomorphi to mtrix lgebr over division lgebr over R. The division lgebrs over R re preisely the rel field R, the omplex field C, nd the quternion skew field H. We sy tht ρ is either rel, omplex, or quternioni. The so-lled Frobenius-Shur inditor provides n esy wy to tell them prt: Theorem 3.1 (Frobenius, Shur). Let be ompt group equipped with its normlized Hr mesure, nd let ρ : L(V ) be n irreduible representtion of. Let χ ρ be the hrter of ρ. Now v(χ ρ ) : χ ρ (g 2 )dg this quntity is lled the Frobenius-Shur inditor. 1 : ρ is rel 0 : ρ is omplex 1 : ρ is quternioni ; (25) In the topologil interprettion from before, the word g 2 orresponds to projetive plne. In ft, we re now prepred to begin onneting the two perspetives. In generl, for group word w in n letters, define the liner funtionl v w (f) : fdµ w. (26) 7

8 The Frobenius-Shur inditor orresponds to the se of w(g) g 2. The funtionl v w, nd thus the mesure µ w, is ompletely determined its vlue t the irreduible hrters: v w (f) χ Ĝ f, χ v w (χ). (27) We do not require tht f be lss funtion beuse µ w is uniform on onjugy lsses. It turns out tht the inditors µ w (χ) lredy tell use wht hppens when we integrte the tul mtries in our representtion, entrywise, without tking tres. Lemm 3.2. Let ρ : L(V ) be n irreduible representtion of ompt group. Then, ρ(w( g))d g v w(χ ρ ) I. (28) dim V n Proof. Set A : ρ(w( g))d g. (29) n A simple hnge of vrible show tht the tion of A ommutes with the tion of : Aρ(h) ρ(w( g))ρ(h)d g n ρ(w( g)h)d g n ρ(hw( h 1 gh))d g n ρ(h) ρ(w( h 1 gh))d g n ρ(h) ρ(w( g ))d g n ρ(h)a, (30) where we hve mde the substitutions g i h 1 g i h nd hve used Hr invrine. We onlude tht A must be n endomorphism of the simple L 2 -module V. By Shur s lemm, A λi, some multiple of the identity. We my ompute λ Tr(A)/ dim V v w (χ ρ )/ dim V. A similr rgument gives us the following: Lemm 3.3. If is ompt group nd ρ : L(V ) is n irreduible representtion, then ρ(xgx 1 )dx χρ (g) I. (31) dim V 8

9 Proof. For g, define I g ρ(xgx 1 )dx. (32) It is esy to see tht the tion of I g ommutes with the tion of : I g ρ(h) ρ(xgx 1 h)dx ρ(hygy 1 )dy ρ(h)i g, (33) where we hve mde the substitution y h 1 x nd hve used Hr invrine. We onlude tht I g must be n endomorphism of the simple L 2 -module V. By Shur s lemm, I g λi, some multiple of the identity. We my ompute λ expliitly: λ Tr(λI) dim V Tr(I g) dim V χρ (xgx 1 )dx dim V χρ (g)dx dim V χρ (g) dim V (34) 4 Surfes nd Witten Zet funtions Theorem 4.1. Let w be word in n letters tht defines surfe (tht is, topologil 2-mnifold), nd let ρ : L(V ) be n irreduible representtion of some ompt group. Now if X(w) is orientble, n ρ(w(t))dt (dim V ) κ 2 I; (35) if not, where κ denotes the Euler hrteristi of X(w). n ρ(w(t))dt v(χ ρ ) 2 κ (dim V ) κ 2 I, (36) 9

10 Proof. We strt with the se of the torus T S 1 S 1 : χ ρ (t 1 t 2 t 1 1 t 1 2 )dt 1 dt 2 Tr ρ(t 1 t 2 t 1 1 t 1 2 )dt 1 dt 2 2 By Lemm 3.2, we in ft hve 2 2 Tr Tr 1 dim V Tr 1 dim V ρ(t 1 t 2 t 1 1 )dt 1 ρ(t 1 2 )dt 2 I t2 ρ(t 1 2 )dt 2 χ ρ (t 2 )ρ(t 1 2 )dt 2 χ ρ (t 2 )χ ρ (t 2 )dt 2 χρ, χ ρ dim V 1 dim V. (37) ρ(t 1 t 2 t 1 1 t 1 2 )dt 1 dt 2 1 I. (38) (dim V ) 2 The lssifition of surfes implies tht n orientble X(w) n be written s onneted sum of tori: X(w) T #T # #T T #k. (39) To form the onneted sum of two words, we simply multiply them in the free produt of their respetive free groups. Erlier propositions llow us to verify the lim for single word representing eh genus. Let us represent the genus γ surfe by the word w(t 1, t 2,..., t 2γ ) [t 1, t 2 ][t 3, t 4 ] [t 2γ 1, t 2γ ]. (40) 10

11 Now we split the T #γ se into γ opies of the T se: ρ([t 1, t 2 ][t 3, t 4 ] [t 2γ 1, t 2γ])dt 2γ ρ([t 1, t 2 ])ρ([t 3, t 4 ]) ρ([t 2γ 1, t 2γ])dt (41) 2γ γ ρ([t 2j 1, t 2j ])dt 2j 1 dt 2j (42) j1 A non-orientble surfe is given by onneted sum of the form γ (dim V ) 2 (43) j1 (dim V ) 2γ (44) X(w) P #T #γ or X(w) P #P #T #γ, (45) where P is projetive plne. It is esy to see, however, tht the theorem of Frobenius nd Shur llows us to pply the sme tehnique in these ses to obtin the required result. Putting this together with erlier lultions (27), we hve Corollry 4.2. For word w with X(w) surfe of Euler hrteristi κ, nd ny mesurble f : C, v w (f) ρ Ĝ f, χ ρ (dim ρ) κ 1 (46) if X(w) is orientble, nd v w (f) ρ Ĝ f, χ ρ v(χ ρ ) 2 κ (dim ρ) κ 1 (47) if X(w) is non-orientble, so long s the sum onverges. Similr formuls were derived by Witten for ompt Lie groups in the ontext of quntum guge theory [Wit91]. Witten goes on to define wht is now known s the Witten zet funtion of ompt group: ζ (s) ρ Ĝ(dim ρ) s. (48) Speil vlues of this zet funtion rise if we plug in the Dir delt funtion tht is, the 11

12 hrter of the regulr representtion for f in the orientble se of Corollry 4.2: v w (δ 1 ) ρ Ĝ δ 1, χ ρ (dim ρ) κ 1 ρ Ĝ χ ρ (1)(dim ρ) κ 1 (dim ρ) κ ρ Ĝ ζ ( κ). (49) This doesn t mke sense beyond the se when is finite. This is fixed by going bkwrds: If ρ Ĝ(dim ρ)κ onverges, the funtion ζ ( κ; g) : ρ Ĝ χ ρ (g)(dim ρ) κ (50) is (s funtion of g) Rdon-Nikodym derivtive for µ w with respet to µ, nd it is ontinuous. So it is the unique 1 ontinuous Rdon-Nikodym derivtive µ w /µ. In the lnguge of distribution theory, ζ (κ; g) is the limit of v w (f j ) over ny sequene f j of L -funtions pproximting point of mss one t g. In prtiulr, t the identity, we hve (µ w /µ)(1) ζ ( κ). (51) In the finite se, we write this formul ombintorilly: 5 Sttistil roup Theory γ (w) n 1 v w (δ 1 ) n 1 ζ ( κ). (52) Returning to our originl exmple of ounting ommuting pirs in Q 8, we need only rell tht the dimensions of the irreduible representtions re {1, 1, 1, 1, 2}. The Witten zet funtion n be written expliitly: ζ Q3 (s) 1 s + 1 s + 1 s + 1 s + 2 s. (53) We reover not only the result tht there re 40 wys to stisfy the ommuttor word Q 8 ζ Q8 (0) 8( ) 40, (54) but lso tht produt of three seprte ommuttors n be stisfied in Q 8 5 ζ Q8 (4) 8 5 ( ) (55) 1 Consider two ontinuous Rdon-Nikodym derivtives. They re equl lmost everywhere. But, if they were unequl t ny point, they would differ t n open set bout tht point, nd nonempty open sets in hve positive mesure. So they re equl. 12

13 different wys. In the introdution, we mentioned severl other wys to hek the degree to whih Q 8 is belin. It is esy to see tht these quntities my lso be lulted in terms of the Witten zet funtion: (1) the index of the derived subgroup: (2) the verge size of onjugy lss: lim ζ Q 8 (s) 4 (56) s Q 8 ζ Q8 (0) 8 5 (57) (3) the dimensions of its lrgest irreduible representtions: ( ) D : lrgest dimension ourring exp lim log(ζ Q 8 (s))/s 2 (58) s number of distint D-dimensionl irreduibles lim s ζ Q 8 (s)/d s 1. (59) We now give n pplition of the theorem in the infinite se. Corollry 5.1. Let SO(3). The word w(t 1, t 2, t 3, t 4 ) [t 1, t 2 ][t 3, t 4 ] provides mesure-preserving mp from the probbility spe 4 to. The resulting mesure on is bsolutely ontinuous with respet to Hr mesure, nd the ontinuous Rdon-Nikodym derivtive t the identity element is π 2 8. (60) Proof. The rottion group SO(3) hs unique irreduible representtions of every odd dimension, nd none of even dimension [BD85]. Thus, n s ζ SO(3) (s) n odd n s n1 (2n) s n1 (1 2 s )ζ(s). (61) So ζ SO(3) (2) (1 2 2 )ζ(2) 3 4 π2 6 π2 8. (62) 13

14 6 A Corollry for Representtions of Finite roups Here we prove the dvertised orollry tht, for finite group, the dimension of n irreduible C-representtion of divides its order. We begin with some esy number theory. Lemm 6.1. Let 1, 2,..., n be integers. In order to show tht d divides eh i, it suffies to verify tht k 1 + k k n 0 (mod d k ) (63) for ll positive integers k. Proof. Suppose first tht d is prime. Let m denote the number of i not divisible by d; we seek to show tht m 0. Setting k ϕ(d n ), Sine ϕ(d n ) d n d n 1 n, ϕ(dn ) 1 + ϕ(dn ) ϕ(dn ) n 0 (mod d ϕ(dn) ). (64) ϕ(dn ) 1 + ϕ(dn ) ϕ(dn ) n 0 (mod d n ). (65) We now pply Euler s totient theorem to eh term on the left; those i whih re prime to p ontribute 1 while the rest re 0. Thus, m 0 (mod d n ). (66) However, m {0, 1,..., n} nd n < d n, so m 0. Consider now the se in whih d is power of some prime p. The ondition for ll positive integers k implies k 1 + k k n 0 (mod d k ) (67) k 1 + k k n 0 (mod p k ) (68) nd so the erlier rgument gives us tht p divides eh i. Dividing through by p, we my iterte this rgument to obtin tht d divides eh i. The generl sttement of the lemm now follows from the Chinese reminder theorem. Corollry 6.2. Let q 1, q 2,..., q n be rtionl numbers. Suppose tht q1 k + q2 k + + qn k Z (69) for eh positive integer k; it follows tht eh q i is n integer. Proof. Cler denomintors nd pply Lemm

15 Theorem 6.3. If is finite group, nd V is n irreduible representtion of over C, then dim V divides. Proof. We pply (52) to n orientble surfe of genus k + 1: For onveniene, define r i to be γ (w k+1 ) 2k+1 ζ (2k) 2k+1 1 (dim ρ) 2k ρ Ĝ ρ Ĝ 2 (dim ρ i ) 2. We now observe: 2k. (70) (dim ρ) 2k γ (w k+1 ) r k 1 + r k r k 1 }{{} + r k 2 + r k r k 2 }{{} + + r k n + r k n + + r k n }{{} (71) Sine the left side is lwys n integer, Corollry 6.2 gives tht r i Z, nd hene the dimension of ny irreduible representtion of divides. Remrk 6.4. Theorem 6.3 relies only on Lemm 3.2, Lemm 3.3, Corollry 4.2, Lemm 6.1, nd Corollry 6.2, the proofs of whih re ompletely elementry if we restrit ttention to finite groups nd words tht re produts of independent ommuttors. In prtiulr, it does not rely on fts bout lgebri integers, Nielsen s lssifition of utomorphisms of free groups, non-belin ohomology, Hr mesure, Rdon-Nikodym derivtives, or the homotopy tegory. Referenes [BD85] [ET68] T. Bröker nd T. Diek. Representtions of Compt Lie roups, volume 98 of rdute Texts in Mthemtis. Springer, P. Erdős nd P. Turán. On some problems of sttistil group theory, iv. At Mth. Hung., 19: , [FRS95] Benjmin Fine, erhrd Rosenberger, nd Mihel Stille. Nielsen trnsformtions nd pplitions: survey. In roups Kore 94, pges Wlter de ruyter, [Nie24] Jkob Nielsen. Die isomorphismengruppe der freien gruppen. Mthemtishe Annlen, 91: , [Wit91] E. Witten. On quntum guge theories in two dimensions. Commun. Mth. Phys., 141(1): ,