On q-analogs of recursions for the number of involutions and prime order elements in symmetric groups

Transcription

1 O -aalogs of recursios for the umber of ivolutios ad prime order elemets i symmetric groups Max B Kutler ad C Rya Viroot Abstract The umber of elemets which suare to the idetity i the symmetric group S is recursive i This recursio may be proved combiatorially, ad there is also a ice expoetial geeratig fuctio for this seuece We study -aalogs of this pheomeo We begi with sums ivolvig -biomial coefficiets which come up aturally whe coutig elemets i fiite classical groups which suare to the idetity, ad we obtai a recursive-lie idetity for the umber of such elemets i fiite special orthogoal groups We the study a -aalog for the umber of elemets i the symmetric group whose pth power is the idetity, for some fixed prime p We fid a Euleria geeratig fuctio for these umbers, ad we prove the -aalog of the recursio for these umbers by givig a combiatorial iterpretatio i terms of vector spaces over fiite fields 2000 Mathematics Subject Classificatio: 05A10, 05A15 Key words ad phrases: -aalogs, -biomial coefficiets, vector spaces over fiite fields, recursios, symmetric group, Euleria geeratig fuctios 1 Itroductio Let S deote the symmetric group o the first positive itegers, ad let T be the umber of elemets g i S such that g 2 is the idetity permutatio Chowla, Herstei, ad Moore [2] studied the seuece {T } (A000085), ad they bega by observig that it is recursive I particular, T 0 = T 1 = 1 (where we let S 0 ad S 1 be trivial groups), ad for 1, T +1 = T + T 1 (11) There is a direct combiatorial argumet which gives (11) Namely, a elemet i S +1 which suares to the idetity either fixes the poit + 1, the umber of which is T, or moves + 1 I the latter case, + 1 ca be paired with oe of the other poits to form a traspositio, which may be combied with a permutatio of the remaiig 1 poits which suares to the idetity, givig T 1 such elemets Aother property of the seuece {T } oticed by Chowla, Herstei, ad Moore [2], is that it has a particularly ice expoetial geeratig fuctio, i that we have =0 T! x = e x+x2 /2 Chowla, Herstei, ad Moore [2] applied the above properties of the umbers T to obtai asymptotic formulas i for T /T 1 ad T, as well as several divisibility properties of T Jacobsthal [8] studied similar properties of the umber of elemets i S of order some fixed 1

2 prime (which we discuss i Sectio 41); Chowla, Herstei, ad Scott [3] gave a expoetial geeratig fuctio for the umber of elemets g of S which satisfy g d = 1 for ay fixed positive iteger d; ad Moser ad Wyma [10] obtaied further results o asymptotic formulas for these uatities More recetly, these ideas have bee applied to study related expoetial geeratig fuctios for larger classes of groups, such as i the papers of Chigira [1] ad Müller [11] The purpose of this paper is to study -aalogs of recursios similar to (11) I Sectio 2, we itroduce properties of the -biomial coefficiets I Sectio 3, we study sums ivolvig -biomial coefficiets which are related to coutig elemets i fiite classical groups which suare to the idetity We prove a recursive-lie formula for these umbers i Theorem 31, ad we apply this to obtai a iterestig result o the umber of order two elemets i fiite special orthogoal groups i Corollary 31 I Sectio 4, we tur to the problem of coutig elemets g i the symmetric group S which satisfy g p = 1, for some fixed prime p (yieldig the seuece A whe p = 5) Sectio 41 summarizes the classical results o this problem I Sectio 42, we give a -aalog for these umbers, for which we give a combiatorial iterpretatio i terms of F -vector spaces (Propositio 41) ad a Euleria geeratig fuctio (Propositio 42) We the prove Theorem 41, a recursive-lie formula for these umbers Fially, i Sectio 43, we specialize to the case p = 2, ad relate our results to other results o ivolutios i the symmetric groups 2 Properties of -biomial coefficiets Let deote a parameter differet from 1 For the sae of combiatorial iterpretatio it will ofte be useful to thi of as the power of some prime For ay positive iteger, we let [] deote the th -iteger, defied as [] = = 1 1 The -iteger [] is a example of a -aalog of a expressio, sice we have lim 1 [] = We defie the -factorial of the iteger 0, deoted []!, recursively as { 1, if = 0; []! = [][ 1]!, if > 0 Oce we have the -factorial, we may defie the -biomial coefficiets i a atural way That is, if ad are o-egative itegers with, we defie ( ) as []! = []![ ]! (21) We recogize the ( -biomial coefficiet as a -aalog of the classical biomial coefficiet sice we have lim 1 ) = ( ) The -biomial coefficiet has a importat combiatorial iterpretatio whe is the power of a prime I this case, ( ) is eual to the umber of -dimesioal subspaces of a - dimesioal vector space over the fiite field F with elemets [13, Propositio 1318] Note that it follows immediately from (21) that we have ( ) ( ) = (22) 2

3 The -biomial coefficiets have may other properties which are very similar to those of the classical biomial coefficiets, ad these properties ca be prove usig the defiitio (21) directly, as well as bijectively i terms of F -vector spaces The followig result is the -Pascal idetity for the -biomial coefficiets, which follows uicly from (21) I a bijective proof of the Propositio 21, the ey observatio is that i a -dimesioal F -vector space V, the umber of -dimesioal subspaces of V which are ot cotaied i some fixed ( 1)-dimesioal subspace of V is exactly ( ) 1 1 Propositio 21 The -biomial coefficiets satisfy the followig property, whe, 1: ( ) ( ) ( ) 1 1 = + 1 It follows from Propositio 21 ad a iductio argumet that the -biomial coefficiets are always polyomials i The ext result is also most directly obtaied by the defiitio (21), although a F -vector space proof may be obtaied by iductio o d Propositio 22 The -biomial coefficiets satisfy the followig property, for ay d 1, ad ay ad such that d: ( ) ( ) d [][ 1] [ d + 1] = [][ 1] [ d + 1] d Fially, defie a -aalog of the expoetial fuctio as e x = j=0 xj [j]! If {A } =0 is ay seuece, the we defie the Euleria geeratig fuctio for the seuece {A } to be the power series A j j=0 [j]! xj That is, a Euleria geeratig fuctio is just a -aalog of a expoetial geeratig fuctio 3 Coutig order 2 elemets i fiite classical groups For the power of a prime, ad 1 a iteger, defie I () = ( ) The followig result is give by Morriso [9, Sectio 111] Propositio 31 If is the power of a odd prime, the I () is the umber of elemets i GL(, F ) which suare to the idetity We ote that Propositio 31 also follows by oticig that a elemet i GL(, F ) which suares to the idetity is determied by a choice of the eigespaces for +1 ad 1, which are complemetary subspaces i F The umber of ways to choose oe of these spaces of dimesio is ( ), ad the umber of complemets is ( ) [12, Lemma 3] If is the power of a odd prime, ad V is a -dimesioal vector space over F, let, be a o-degeerate symmetric form o V The special orthogoal group for, is defied to be the group of ivertible F -liear trasformatios of V with determiat 1 which preserve the form, See [7], for example, for the classificatio of symmetric biliear forms over F, ad the isomorphism classes of the correspodig special orthogoal groups I particular, whe is odd, there is exactly oe isomorphism class of special orthogoal group, which we deote 3

4 by SO(, F ), ad whe is eve, there are two classes of these groups, which we deote by SO + (, F ) ad SO (, F ), or by SO ± (, F ) to deote either class of group simultaeously The secod-amed author [14, Sectio 4] proved the followig, usig results o the cetralizers of cojugacy classes i the fiite special orthogoal groups Propositio 32 Let be the power of a odd prime If G = SO ± (2m, F ), the the umber of elemets i G which suare to the idetity is I m ( 2 ) If G = SO(2m + 1, F ), the the umber of elemets i G which suare to the idetity is m ( ) m 2(m +1) Propositio 32 prompts the followig defiitio geeralizig I () For ay iteger j, defie I (j) () by I (j) () = ( +j) So, I (0) () = I (), ad the umber of elemets i SO(2m + 1, F ) which suare to the idetity is I m (1) ( 2 ) The umbers I (j) () have the followig property Theorem 31 The umbers I (j) () satisfy the formula I (j) +1 () = I(j) Proof First, by Propositio 21, we have () + j ( 1)I (1 j) 1 =1 2 () + (+1)j I (2 j) () I (j) () = ( ) ( ( ) (+1 +j) = 1 + (+1 +j) We may rewrite this as ( ) ) ( +j) + ( (+1 +j) ( +j) ) + (+1)(+1 )+j 1 =1 = I (j) () + ( +j) ( 1) + (+2)( )+(+1)j Now, by applyig Propositio 22 with d = 1, ad the reidexig the first summatio, we have I (j) +1 () = I(j) () + ( 1) =1 1 ( +j) ( 1 = I (j) () + ( 1) (+1)( 1+j) (+2)( )+(+1)j ) + (+2)( )+(+1)j Fially, by (22), we may switch with 1 i the first summatio, ad with i the secod summatio above to obtai I (j) +1 () = I(j) 1 () + ( 1) 1 ( )(+j) + 4 ( +2)+( +1)j

5 1 1 = I (j) () + ( 1) j ( j) + = I (j) () + j ( 1)I (1 j) 1 (+1)j ( +2 j) () + (+1)j I (2 j) (), as desired If we let j = 0 i Theorem 31, the we have the followig relatioship betwee the umber of elemets i GL( + 1, F ) ad i GL(, F ) which suare to the idetity, whe is the power of a odd prime: I +1 () = I () + ( 1)I (1) 1 () + I(2) This is remiiscet of the recursio of the umbers G () = (, which satisfy G +1() = 2G () + ( 1)G 1 (), a result due to Goldma ad Rota [6] If we let j = 1 i Theorem 31, ad replace with 2, where is the power of a odd prime, the we get the followig particularly ice result, which follows from Propositio 32 Corollary 31 Let be the power of a odd prime, ad let > 1 The umber of elemets which suare to the idetity i SO(2 + 3, F ) (which is I (1) +1 (2 )), i SO(2 + 1, F ) (which is I (1) ( 2 )), ad i SO(2 2, F ) (which is I (0) 1 (2 )) are related i the followig way: () ) I (1) +1 (2 ) = ( )I (1) ( 2 ) + 2 ( 2 1)I (0) 1 (2 ) It would be iterestig to have a purely combiatorial proof of Corollary 31, by directly coutig elemets which suare to the idetity i these groups, but we have ot bee able to fid such a proof 4 A -aalog of the umber of prime order elemets i S 41 Coutig prime order elemets i S Let p be prime, ad let T be the umber of elemets g i S such that g p is the idetity, ad here we give the I the Itroductio, we gave some properties of the umbers T = T (2) aalogous properties of T due to Jacobsthal [8] A elemet g i S +1 which satisfies g p = 1 either fixes the poit +1, or it does ot There are T such elemets i S +1 which fix the poit + 1 If such a elemet does ot fix the poit + 1, the it cotais a p-cycle permutig + 1 The umber of such p-cycles ca be see to be ( 1) ( p + 2) by assumig that, i cycle otatio, + 1 appears i the first positio, ad the choosig the remaiig p 1 poits i the cycle from the set {1, 2,, } To costruct the rest of the elemet, we may permute the remaiig p + 1 poits i such a way that our elemet still satisfies g p = 1, ad the umber of ways i which to do this is exactly T p+1 This implies that we have the followig recursio: T +1 = T + ( 1) ( p + 2)T p+1 (41) We may also eumerate T directly A elemet g i S satisfyig g p = 1 may cosist of aywhere from 0 (the idetity elemet) to /p disjoit p-cycles If g cosists of exactly disjoit p-cycles, the we may choose the poits permuted i these p-cycles i ( p) ways We the order these poits i oe of! ways, where each bloc of p poits form a p-cycle But 5

6 each of these p-cycles, writte i cycle otatio, may be writte i p ways, ad the differet p- cycles may be ordered i ay of! ways to obtai the same elemet Puttig all of this together, we obtai the formula /p T =! p! (42) p We may rewrite (42) i a few differet useful ways For ay iteger m 2, we defie the m-fold multifactorial of a iteger r 0, deoted r! (m) as the product r! (m) = r(r m)(r 2m) (r tm), where t 0 is maximum such that r > tm So, for example, r! (2), which is also writte r!!, is the product of every other decreasig positive iteger, startig with r Usig this otatio, ad by rewritig the biomial coefficiets as factorials ad reidexig, we may write T as /p T =!! = p i,j 0,pi+j=! p i i!j! (43) It follows from the secod formula for T geeratig fuctio for T i (43) above, that e x+xp /p is the expoetial 42 A -aalog For ay iteger m 2, ad a parameter, defie the m-fold -multifactorial of a iteger r 0, deoted [r]! (m), by [r]! (m) = [r][r m][r 2m] [r tm], where t 0 is maximum such that r > tm Cotiue to let p be a prime, ad let 0 Defie T () to be the followig: /p T () = It is immediate from the first formula for T that T () is a -aalog of T [p]! [p]! (44) p i (43) that we have lim 1 T () = T, so I the case that is the power of a prime (idepedet of the prime p), there is a combiatorial iterpretatio of T () Before that, we eed a defiitio If V is ay vector space of dimesio, defie a complete flag of V to be a chai of subspaces of V, {0} = V 0 V 1 V 1 V = V, such that dim(v i ) = i for each i For example, if we have a basis {v 1, v 2,, v } of V, the a complete flag of V may be obtaied by taig V i = spa{v 1,, v i } Heceforth, we will let a lie i a vector space V mea a oe-dimesioal subspace of V Propositio 41 Let p be prime, the power of a prime (idepedet of p), 0, ad V a -dimesioal F -vector space The T () is the umber of ways to choose a subspace W of V with dimesio divisible by p, together with, for some complete flag of W, a seuece of lies, oe i each subspace of the complete flag with dimesio ot divisible by p 6

7 Proof First, to choose a subspace of V with dimesio divisible by p, we must tae some oegative iteger /p, ad choose a subspace W of V of dimesio p, the umber of which is ( ) p Now, give a complete flag of W, if W i is a subspace i the flag with dim(w i ) = i, the the umber of lies i W i is ( ) i 1 = [i] So, the umber of ways to choose a seuece of lies, oe i each subspace of dimesio ot divisible by p i a give flag of W, is exactly [p 1][p 2] [p p + 1][p p 1][p p 2] [p + 1][p 1] [2][1] = Puttig all of this together, we get exactly the formula for T () i (44) [p]! [p]! We ca also describe T () i terms of a Euleria geeratig fuctio, as follows Propositio 42 The Euleria geeratig fuctio for T () is give by ( ) x pi [pi]! e x i=0 Proof We have ( ) x pi [pi]! e x x pi x j /p = [pi]! [j]! = 1 [p]! x [ p]! i=0 i=0 j=0 =0 /p = [p]! []! x [p]! [p]![ p]! []! = T () x, []! as desired =0 Next, we see that the T () have a property similar to the recursio i (41) of the T Theorem 41 For p a prime, ad 0, defie R () to be the followig polyomial: /p R () = The for all p 1, the T () satisfy T +1 ad T () = 1 for 0 p 2 p [p]! [p]! p =0 () = T () + [][ 1] [ p + 2]R p+1 (), Note that if we let 1, the idetity for the T () i Theorem 41 turs ito the recursio (41), sice we have lim 1 R usig Propositio 41 () = T We ow give a combiatorial proof of Theorem 41 Proof of Theorem 41 Let V be a ( + 1)-dimesioal F -vector space, ad let {v 1,, v +1 } be a basis for V By Propositio 41, T +1 () is the umber of ways to choose a subspace W of V with dimesio divisible by p, together with a lie i each of the subspaces i some complete flag of W with dimesio ot divisible by p We ow cout this uatity aother way 7

8 Defie V 0 = {0} ad V i = spa{v 1,, v i } for 1 i + 1 For ay subspace W of V of dimesio divisible by p, say dim(w ) = p, we tae a complete flag of W as follows Let W 0 = W V 0 = {0}, ad for each j such that 1 j p, defie W j = W V i, where i is miimal so that dim(w j ) = j Now, W 0 W 1 W p = W is a complete flag of W First suppose that the subspace W is cotaied i V The umber of ways to choose a subspace W of V of dimesio divisible by p, alog with a seuece of lies i each space of dimesio ot divisible by p i the complete flag of W defied above, is eual to T () by Propositio 41 Now cosider the case that our subspace W with dimesio divisible by p is ot a subspace of V Suppose that dim(w ) = p, where satisfies 1 p Sice W is ot completely cotaied i V, we must have dim(w V ) = p 1, ad so W V = W p 1 By the remars give before Propositio 21, the umber of ways to choose our subspace W is +1 p( ) p 1 Now that we have chose our subspace W, we ow choose lies from the subspaces W i, where p p+1 i p 1 The umber of ways to do this is exactly [p p+1][p p+2] [p 1] So ow, the umber of ways to select W, together with these lies, is +1 p [p 1] [p p + 2][p p + 1] p 1 where [p 1] [p p+2][p p+1] ( ) p 1 couts the umber of ways to select our W = W p 1, together with these lies By Propositio 22 with d = p 1, we ow that this is also couted by [][ 1] [ p + 2] ( ) p+1 p p, by first choosig lies from V i, for p + 2 i So, the total umber of ways to select our space W, together with these p 1 lies, is ) ( p p [][ 1] [ p + 2] p p Now, cosider the complete flag {0} = W 0 W p = W of W, where {0} = W 0 W p p is a complete flag of W p p We have already selected lies from the subspaces W p p+1,, W p 1, so we eed oly choose lies from the remaiig W i with dimesio ot divisible by p By the argumet i the proof of Propositio 41, the umber of ways to do this is [p p]! Puttig all of this together, the umber of ways to choose a dimesio p subspace W [p p]! of V which is ot cotaied i V, together with a seuece of lies, oe each from the subspaces of dimesio ot divisible by p i our complete flag of W, is ( ) [][ 1] [ p + 2] +1 p [p p]! p + 1 [p p]! p p Summig over all possible ad reidexig, gives a total of [][ 1] [ p + 2]R p+1 () We ote that it follows from this argumet that R p+1 () is the umber of ways, after choosig lies from V i for p+2 i, to choose our subspace ad the remaiig lies appropriately Addig [][ 1] [ p + 2]R p+1 () to T (), the cout for the case where W is cotaied i V, gives the desired result 43 The case p = 2 We bega i the Itroductio with a discussio of the umber of elemets which suare to the idetity, or ivolutios, i the symmetric group The set of ivolutios of the symmetric group, 8

9 is of particular iterest, because, for example, the cetralizer of a fixed poit free ivolutio is a Weyl group of type B, ad plays a importat role i Lie theory There are various statistics o ivolutios which are useful to eep trac of usig geeratig fuctios [4, 5] Such geeratig fuctios ca provide for more examples of -aalogs of the umber T (2) of ivolutios i S Deodhar ad Sriivasa [4], for example, defie the weight of a ivolutio δ S, deoted wt(δ), as follows If δ is writte i cycle otatio, ad (i j) is oe of the disjoit traspositios of δ, the write (i j) δ with the assumptio i < j, ad defie sp(i j) = j i 1 If we draw a arc diagram for δ, where a arc from i to j deotes (i j) δ, the let c(δ) deote the umber of crossigs of arcs i the diagram (see [4, Sectio 1]) Now defie the weight of δ as wt(δ) = sp(i j) c(δ) (i j) δ Let Iv(, ) deote the set of ivolutios i S which are a product of exactly disjoit traspositios, ad defie i (, ) to be the geeratig fuctio i (, ) = wt(δ) δ Iv(,) Deodhar ad Sriivasa [4, Propositio 21] proved that the i (, ) satisfy the recursio i ( + 1, ) = i (, ) + []i ( 1, 1) (45) If we defie I () = /2 i (, ), the it follows from (45) that I +1 () = I () + []I 1 (), (46) which is a -aalog of (11) Ideed, it follows from the defiitio of i (, ) as a geeratig fuctio that lim 1 I () = T (2) It is uclear what is exactly the coectio betwee I () ad T (2) (), or whether I () has a iterpretatio i terms of F -vector spaces Such a iterpretatio seems possible, sice i (2, ) = [2]!/[2]!! = [2 1]!! for 1 [4, Corollary 22], while /2 T (2) () = [2]! [2]!! ( ) /2 = =1 [2 1]!! 2 We might expect a coveiet Euleria geeratig fuctio f(x) for I () Based o Propositio 42, we mae the guess that f(x) has the form f(x) = F (x)e x for some power series F (x) = m=0 a x We may i fact solve for the a from euatios obtaied from f(x) = F (x)e x, startig with a 0 = I 0 () = 1, by computig the values of I () Calculatig the first few values of a, we fid F (x) = [2]! x x x 4 + [3]! [4]! Ufortuately, this does ot seem to shed ay light o the meaig of I () i terms of F -vector spaces Liewise, we also do ot ow if T (2) () has a iterpretatio as a geeratig fuctio for ivolutios i S with respect to some statistic It would be satisfyig to uderstad ay coectios amogst these poits of view We coclude by remarig that it would be of iterest to have a statistic o order p elemets of S to geeralize the wor of Deodhar ad Sriivasa [4], ad to uderstad the geeral coectio betwee T () ad statistics o order p elemets i symmetric groups 9

10 5 Acowledgmets The first-amed author was supported by NSF grat DMS , at the College of William ad Mary REU The secod-amed author was supported by a summer research grat from the College of William ad Mary Both authors tha Shahla Nasserasr for several helpful discussios, ad the aoymous referee for suggestios to improve this paper Refereces [1] N Chigira, The solutios of x d = 1 i fiite groups, J Algebra 180 (1996), [2] S Chowla, I N Herstei, ad W K Moore, O recursios coected with symmetric groups I, Caad J Math 3 (1951), [3] S Chowla, I N Herstei, ad W R Scott, The solutios of x d = 1 i symmetric groups, Norse Vid Sels Forh, Trodheim 25 (1952), [4] R S Deodhar ad M K Sriivasa, A statistic o ivolutios, J Algebraic Combi 13 (2001), [5] W M B Dues, Permutatio statistics o ivolutios, Europea J Combi 28 (2007), [6] J Goldma, G-C Rota, The umber of subspaces of a vector space, i Recet Progress i Combiatorics (Proc Third Waterloo Cof o Combiatorics, 1968), Academic Press, 1969, pp [7] L C Grove, Classical Groups ad Geometric Algebra, Graduate Studies i Mathematics, 39, America Mathematical Society, 2002 [8] E Jacobsthal, Sur le ombre d élémets du groupe symétriue S dot l ordre est u ombre premier, Norse Vid Sels Forh, Trodheim 21 (1949), [9] K Morriso, Iteger seueces ad matrices over fiite fields, J Iteger Se 9 (2006), Article 0621, 28 pp [10] L Moser ad M Wyma, O the solutios of x d = 1 i symmetric groups, Caad J Math 7 (1955), [11] T Müller, Eumeratig represetatios i fiite wreath products, Adv Math 153 (2000), [12] M K Sriivasa, The Euleria geeratig fuctio of -deragemets, Discrete Math 306 (2006), [13] R P Staley, Eumerative Combiatorics, Vol 1, Cambridge Studies i Advaced Mathematics, 49, Cambridge Uiversity Press, 1997 [14] C R Viroot, Character degree sums ad real represetatios of fiite classical groups of odd characteristic, to appear i J Algebra Appl, 10

11 (Cocered with seueces A ad A052501) Departmet of Mathematics Harvey Mudd College 301 Platt Boulevard Claremot, CA Departmet of Mathematics College of William ad Mary P O Box 8795 Williamsburg, VA viroot@mathwmedu 11