An enumeration of flags in finite vector spaces

Transcription

1 A eumeratio of flags i fiite vector spaces C Rya Viroot Departmet of Mathematics College of William ad Mary P O Box 8795 Williamsburg VA viroot@mathwmedu Submitted: Feb ; Accepted: Ju ; Published: Jul Mathematics Subject Classificatios: 05A19 05A15 05A30 Abstract By coutig flags i fiite vector spaces we obtai a -multiomial aalog of a recursio for -biomial coefficiets proved by Nijehuis Solow ad Wilf We use the idetity to give a combiatorial proof of a ow recurrece for the geeralized Galois umbers 1 Itroductio For a parameter 1 ad a positive iteger let ad 0 1 For o-egative itegers ad with the -biomial coefficiet or Gaussia polyomial deoted is defied as The Rogers-Szegö polyomial i a sigle variable deoted H t is defied as H t 0 t The Rogers-Szegö polyomials first appeared i papers of Rogers [16 17] which led up to the famous Rogers-Ramauja idetities ad later were idepedetly studied by Szegö [19] They are importat i combiatorial umber theory [1 Ex 33 39] ad [5 Sec 20] symmetric fuctio theory [20] ad are ey examples of orthogoal polyomials [2] They also have applicatios i mathematical physics [11 13] The Rogers-Szegö polyomials satisfy the recursio see [1 p 49] Supported by NSF grat DMS H +1 t 1 + th t + t 1H 1 t 11 the electroic joural of combiatorics #P5 1

2 Lettig t 1 we have H 1 0 which whe is the power of a prime is the total umber of subspaces of a -dimesioal vector space over a field with elemets The umbers G H 1 are the Galois umbers ad from 11 satisfy the recursio G +1 2G + 1G 1 12 The Galois umbers were studied from the poit of view of fiite vector spaces by Goldma ad Rota [6] ad have bee studied extesively elsewhere for example i [15 9] I particular Nijehuis Solow ad Wilf [15] give a bijective proof of the recursio 12 usig fiite vector spaces by provig for itegers For o-egative itegers 1 2 m such that m we defie the -multiomial coefficiet of legth m as 1 2 m 1 2 m so that If deotes the m-tuple 1 m write the correspodig -multiomial coefficiet as For a subset J {1 m} let e 1 m J deote the m-tuple e 1 e m where { 1 if i J e i 0 if i J For example if m 3 J {1 3} ad the The e J mai result of this paper which is obtaied i Sectio 2 Theorem 21 is a combiatorial proof through eumeratig flags i fiite vectors spaces of the followig geeralizatio of the idetity 13 For m 2 ad ay 1 m > 0 such that m + 1 we have J J 1 m J +1 e J J {1m} J >0 I Sectio 3 we prove a recursio which geeralizes 12 I particular the geeralized Galois umber G m is defied as G m 1 2 m m which i the case that is the power of a prime eumerates the total umber of flags of legth m 1 of a -dimesioal F -vector space Quite recetly the asymptotic statistics of these geeralized Galois umbers have bee studied by Bliem ad Kousidis [3] ad Kousidis [12] the electroic joural of combiatorics #P5 2

3 Directly followig from Theorem 21 we prove i Theorem 31 that for m 1 G m +1 i0 m i i G m i which also follows from a ow recurrece for the multivariate Rogers-Szegö polyomials i 2 Flags i fiite vector spaces I this sectio let be the power of a prime ad let F deote a fiite field with elemets If V is a -dimesioal vector space over F the the -biomial coefficiet is the umber of -dimesioal subspaces of V see [10 Thm 71] or [18 Prop 1318] So the Galois umber G H 1 0 is the total umber of subspaces of a -dimesioal vector space over F Now cosider the -multiomial coefficiet i terms of vector spaces over F It follows from the defiitio of a -multiomial coefficiet ad the fact that that we have 1 1 m m m m 2 So if V is a -dimesioal vector space over F the -multiomial coefficiet 1 m is eual to the umber of ways to choose a 1 -dimesioal subspace W 1 of V a 1 2 -dimesioal subspace W 2 of W 1 ad so o util fially we choose a 1 -dimesioal subspace W of some 1 m 2 -dimesioal subspace W m 2 see also [14 Sec 15] That is W W m 2 W 2 W 1 is a flag of subspaces of V of legth m 1 where dim W i i j1 j We ow tur to a bijective proof of the idetity 13 that for itegers While the bijective iterpretatio of this idetity which we give ow is differet from the proof give by Nijehuis Solow ad Wilf i [15] it is the iterpretatio which is most the electroic joural of combiatorics #P5 3

4 helpful for the proof of our mai result Fix V to be a + 1-dimesioal F -vector space There are +1 ways to choose a -dimesioal subspace W of V Fix a basis {v 1 v 2 v +1 } of V Ay -dimesioal subspace W ca be writte as spaw v where W is a 1-dimesioal subspace of V spav 1 v for some v We may choose W i three distict ways If v V the W is a subspace of V for which there are choices Call this a type 1 subspace of V If v +1 W the we may tae v v +1 ad W is determied by W for which there are choices We call this a 1 type 2 subspace of V Fially if both W V ad v +1 W the we call W a type 3 subspace of V ad it follows from 13 ad ca be show directly as well that there are 1 1 choices for W 1 We may ow prove our mai result Theorem 21 For m 2 ad ay 1 m > 0 such that m + 1 we have J J 1 m J +1 e J J {1m} J >0 Proof Fix V to be a + 1-dimesioal vector space over F Fix a basis of each subspace U of V so that we may spea of subspaces of type 1 2 or 3 of each subspace U with respect to this fixed basis Cosider a flag F of subspaces of V W 0 W W 2 W 1 such that if we defie i for 1 i m by i j1 j + 1 dim W i the each i > 0 The total umber of such flags is +1 1 m Cosider ow a labelig of such flags i the followig way Give a flag F as above defie ad r mi{1 j m W j is a type 1 subspace of W j 1 } J {r} {1 j r 1 W j is a type 3 subspace of W j 1 } Defie the flag F to be a type J flag of V That is for ay oempty J {1 m} we may spea of flags of type J of V We shall prove that 1 J J 21 J +1 e J is the umber of type J flags of legth m 1 of the F -space V Oce this claim is prove we will have accouted for all 2 m 1 terms o the right-side of the desired result of Theorem 21 ad all possible ways to choose our flag We prove the claim by iductio o m where the base case of m 2 follows from 13 ad its iterpretatio i terms of subspaces of types 1 2 ad 3 as give above We must cosider each possible oempty J {1 m} ad show that i each case the uatity 21 couts the umber of type J flags So cosider a flag of subspaces W W 2 W 1 of V where dim W i + 1 i j1 j First if J {1} the the umber of ways to choose W 1 to be a type 1 subspace of V of dimesio +1 1 is while the umber of ways to choose the remaiig legth +1 1 the electroic joural of combiatorics #P5 4

5 m 2 flag W W 2 of W 1 is exactly m Thus the total umber of ways to choose our flag of type J with J {1} is m m which is exactly the expressio 21 for J {1} as claimed So we ow suppose J {1} so if r is the maximum elemet of J we have r > 1 We cosider the cases of whether 1 J or 1 J separately Suppose that 1 J The we must choose our flag so that W 1 is a type 2 subspace of V of which there are 1 such subspaces Now if we defie I J 1 {j 1 j J} so that I {1 m 1} ad I J we must choose the rest of our type J flag of V by choosig a type I flag of W 1 of legth m 2 If we let 2 m the by our iductio hypothesis the umber of type I flags of legth m 2 of the dimesioal space W 1 is 1 I I I e I So the total umber of ways to choose the type J flag of legth m 1 i V is I 1 A direct computatio yields 1 1 I I I +1 ad further ote that + 1 I I 1 I + 1 e I e I + 1 I I +1 1 I + 1 where 1 m Together these give I 1 1 I I J e J e I 1 J J J +1 e J givig the claim that whe 1 J the umber of type J subspaces of legth m 1 of V is give by 21 Fially suppose that 1 J ad J {1} So we must choose our flag so that W 1 is a type 3 subspace of V ad there are such subspaces If we let I J 1 \ {0} so that ow J I + 1 the we must choose the rest of our flag as a type I flag of legth m 2 of W 1 Lettig agai 2 m the by our iductio hypothesis the total umber of flags of type J of legth m 1 of V is give by I I 1 1 I +1 e I the electroic joural of combiatorics #P5 5

6 A computatio gives I +1 1 I I 1 I + 1 ad also ote I I 1 I + 1 e I + 1 J e J where 1 m sice I J 1 We fially obtai that I I 1 1 I +1 e I 1 J 1 J +1 is the the umber of type J subspaces of legth m 1 of V as claimed 3 Geeralized Galois umbers + 1 J e J Defie the homogeeous Rogers-Szegö polyomial i m variables for m 2 deoted H t 1 t 2 t m by H t 1 t 2 t m t 1 1 t m m 1 m m ad defie the Rogers-Szegö polyomial i m 1 variables deoted H t 1 t by H t 1 t Ht 1 t 1 The homogeeous multivariate Rogers-Szegö polyomials were first defied by Rogers [16] i terms of their geeratig fuctio ad several of their properties are give by Fie [5 Sectio 21] The defiitio of the multivariate Rogers-Szegö polyomial H is give by Adrews i [1 Chap 3 Ex 17] alog with a geeratig fuctio although there is little other study of these polyomials elsewhere i the literature however there is a o-symmetric versio of a bivariate Rogers-Szegö polyomial [4] The multivariate Rogers-Szegö polyomials satisfy a recursio which geeralizes 11 although it seems ot to be very well-ow as the oly proof ad referece to it that the author has foud is i the physics literature i papers of Hiami [7 8] For ay fiite set of variables X let e i X deote the ith elemetary symmetric polyomial i the variables X The the Rogers-Szegö polyomials i m 1 variables satisfy the followig recursio: H +1 t 1 t i0 e i+1 t 1 t 1 1 i H i t 1 t 31 i the electroic joural of combiatorics #P5 6

7 G m The sum of all -multiomial coefficiets of legth m or the geeralized Galois umber is the H G m 1 m m From the discussio at the begiig of Sectio 2 whe is the power of a prime G m is exactly the total umber of flags of subspaces of legth m 1 i a -dimesioal F -vector space Sice the umber of terms i the elemetary symmetric polyomial e i+1 t 1 t 1 is m i+1 the the followig our last result follows directly from the formal idetity 31 proved by Hiami However we give a proof which follows directly from Theorem 21 ad is thus a bijective proof through the eumeratio of flags i a fiite vector space Theorem 31 The geeralized Galois umbers satisfy the recursio for m 1 G m +1 i0 m i i G m i Proof For coveiece wheever ay i < 0 we defie the -multiomial coefficiet 1 2 m 0 Gratig this we have Theorem 21 holds for all i 0 We ow begi with the defiitio of G m +1 as the sum of all -multiomial coefficiets ad we directly apply Theorem 21 to rewrite the sum as follows: G m m 1 + m+1 1 J J 1 + m+1 J {1m} J >0 i0 i0 i0 J {1m} J >0 1 m 1 + +m+1 J {1m} J i+1 m i + 1 m i m 1 + +m m i 1 m J +1 1 J 1 J +1 1 i i 1 1 i G m i i i i e J + 1 J e J i e J i where the ext-to-last euality follows from the fact that each idex may be obtaied from a idex from ay of the m i+1 subsets J of size i + 1 i the electroic joural of combiatorics #P5 7

8 By a very similar argumet we may see that i fact the recursio for the multiomial Rogers-Szegö polyomials i 31 also follows from Theorem 21 Acowledgemets The author thas George Adrews ad Ket Morriso for very helpful commets ad the aoymous referee for very useful suggestios to improve this paper Refereces [1] G Adrews The Theory of Partitios Ecyclopedia of Mathematics ad its Applicatios Addiso-Wesley Readig Mass-Lodo-Amsterdam 1976 [2] R Asey ad J Wilso Some basic hypergeometric orthogoal polyomials that geeralize Jacobi polyomials Mem Amer Math Soc [3] T Bliem ad S Kousidis The umber of flags i fiite vector spaces: asymptotic ormality ad Mahoia statistics J Algebraic Combi to appear doi:101007/s [4] W Y C Che H L Saad ad L H Su The bivariate Rogers-Szegö polyomials J Phys A 4023: [5] N J Fie Basic Hypergeometric Series ad Applicatios Mathematical Surveys ad Moographs 27 America Mathematical Society Providece RI 1988 [6] J Goldma ad G-C Rota The umber of subspaces of a vector space I Recet Progress i Combiatorics Proc Third Waterloo Cof o Combiatorics 1968 pages Academic Press New Yor 1969 [7] K Hiami Represetatios of motifs: ew aspect of the Rogers-Szegö polyomials J Phys Soc Japa 644: [8] K Hiami Represetatio of the Yagia ivariat motif ad the Macdoald polyomial J Phys A 307: [9] S Hitzema ad W Hochstättler O the combiatorics of Galois umbers Discrete Math 31024: [10] V Kac ad P Cheug Quatum Calculus Uiversitext Spriger-Verlag New Yor 2002 [11] H Karabulut Distributed Gaussia polyomials as -oscillator eigefuctios J Math Phys 471: [12] S Kousidis Asymptotics of geeralized Galois umbers via affie Kac-Moody algebras Proc Amer Math Soc to appear arxiv: [13] A J Macfarlae O -aalogues of the uatum harmoic oscillator ad the uatum group SU2 J Phys A 2221: [14] K Morriso Iteger seueces ad matrices over fiite fields J Iteger Se 92: the electroic joural of combiatorics #P5 8

9 [15] A Nijehuis A E Solow ad H S Wilf Bijective methods i the theory of fiite vector spaces J Combi Theory Ser A 371: [16] L J Rogers O a three-fold symmetry i the elemets of Heie s series Proc Lodo Math Soc 24: [17] L J Rogers O the expasio of some ifiite products Proc Lodo Math Soc 24: [18] R P Staley Eumerative Combiatorics Vol 1 Cambridge Studies i Advaced Mathematics 49 Cambridge Uiversity Press Cambridge 1997 [19] G Szegö Ei Beitrag zur Theorie der Thetafutioe S B Preuss Aad Wiss Phys-Math Kl [20] S O Waraar Rogers-Szegö polyomials ad Hall-Littlewood symmetric fuctios J Algebra 3032: the electroic joural of combiatorics #P5 9