The Rogers-Ramanujan Identities, the Finite General Linear Groups, and the Hall-Littlewood

a r X i v :m a t h /9712236v 1 [m a t h .C O ] 9 D e c 1997The Rogers-Ramanujan Identities,the Finite General Linear Groups,and the Hall-LittlewoodPolynomialsBy Jason Fulman Dartmouth CollegeJason.E.Fulman@AbstractThe Rogers-Ramanujan identities have been studied from the viewpoints of combinatorics,number theory,affine Lie algebras,statistical mechanics,and quantum field theory.This note connects the Rogers-Ramanujan identities with the finite general linear groups and the Hall-Littlewood polynomials of symmetric function theory.1IntroductionThe Rogers-Ramanujan identities are among the most famous partition identities in all of number theory and combinatorics.This note will be concerned with the following generalization of the Rogers-Ramanujan identities,due to Gordon.Let (x )n denote (1−x )(1−x 2)···(1−x n ).Theorem 1(Andrews [1],page 111)For 1≤i ≤k,k ≥2,and |x |<1n 1,···,n k −1≥0x N 21+···+N 2k −1+N i +···+N k −11−x rwhere N j =n j +···n k −1.Gordon’s generalization of the Rogers-Ramanujan identities have been widely studied and ap-pear in many places in mathematics and physics.Andrews [1]discusses combinatorial aspects of these identities.Berndt [4]describes some number theoretic connections.Feigen and Frenkel [8]interpret the Gordon identities as a character formula for the Virasoro algebra.Andrews,Baxter,and Forrester [2]relate Gordon’s generalization to exactly solved models in statistical mechanics.Berkovich,McCoy,and Orrick [3]and Kirillov [11],[12]discuss the connection with quantum field theory.This note studies the i =k case of Gordon’s generalization of the Rogers-Ramanujan identities.Section 2describes the relation with the finite general linear groups.Section 3describes the relation with symmetric function theory.The results here are taken from the Ph.D.thesis of Fulman [5].We use the following standard notation from the theory of partitions.λis said to be a partitionof n =|λ|if λ1≥λ2≥···≥0andi λi =n .The λi are referred to as the parts of λ.Let m i (λ)be the number of parts of λof size i ,and define λ′i =m i (λ)+m i +1(λ)+···.By n (λ)is meanti ≥1(i −1)λi .2Relation with the Finite General Linear GroupsRecall from elementary linear algebra (for instance from Chapter 6of Herstein [9])that the con-jugacy classes of GL (n,q )are parameterized by rational canonical form.This form corresponds to the following combinatorial data.To each monic non-constant irreducible polynomial φover a field of q elements,associate a partition (perhaps the trivial partition)λφof some non-negative integer |λφ|.Let m φdenote the degree of φ.The only restrictions necessary for this data to represent a conjugacy class are:1.|λz |=02.φ|λφ|m φ=nDefinition For α∈GL (n,q )and φa monic,irreducible polynomial over F q ,a field of q elements,define λφ(α)to be the partition corresponding to the polynomial φin the rational canonical form of α.The following elementary lemmas will be of use.Lemma 1If f (1)<∞and f has a Taylor series around 0,thenlim n →∞[u n ]f (u )1−u=ni =0a i .2Lemma 2is proved by Stong [17](using the fact that there are q n (n −1)unipotent elements in GL (n,q )),but we give a simpler proof.Lemma 21−u =φ=z ∞r =1(1−u m φqm φt )=1−uq t −1.Expanding1q mφas ageometric series,the coefficient of u d in the reciprocal of the left hand side is1qrm φ)=r ≥1(1−uq m φr)Proof:Assume for simplicity of notation that φ=z −1.From the proof it will be clear that the general case follows.Stong [17],using Kung’s [13]formula for the sizes of the conjugacy classes of GL (n,q ),estab-lished the following “cycle index”for the general linear groups:1+∞ n =1u nim ik =1(qm φd i −q m φ(d i −k ))]where the product is over all monic φ=z which are irreducible polynomials over the field of q elements,andd i (λ)=1m 1(λ)+2m 2(λ)+···+(i −1)m i −1(λ)+(m i (λ)+m i +1(λ)+···+m j (λ))i Observe that:i m ik =1(qm φd i−qm φ(d i −k ))=qm φim i (λ)d i (λ)i(1q m φ)m i (λ)=qm φi[im i (λ)2+2m i (λ)h<ihm h (λ)]i(1q m φ)m i (λ)Combining this observation with Lemma 2shows that:1+∞ n =1u n1−u φ=z ∞r =1(1−u m φqm φi(λ′i )2i (11−u (∞r =1(1−uqi(λ′i )2i (1qr))(λ:λ1<k1q )m i (λ))=∞r =1r =0,±k (mod 2k +1)(1−1q .2Remarks1.Recall that an α∈Mat (n,q ),all n ∗n matrices with entries in F q ,is said to be semisimpleif it is diagonalizable over ¯Fq ,the algebraic closure of F q .It is elementary to show that αissemisimpleif and only if all λφ(α)have largest part at most bining this with Theorem 2and Stong’s cycle index for Mat (n,q )one can prove that the n →∞limiting probability that an element of Mat (n,q )is semisimple is:∞r =1r =0,±2(mod 5)(1−1(1−x )(1−x 2)···(1−x n )=∞ m =11(1−x )(1−x 2)···(1−x n )=∞ m =11q )∞ n =01|AGL (n,q )|where AGL (n,q )is the affine finite general linear group.This suggests that an analog ofTheorem 2should exist for the affine finite general linear groups.4.Is there some group theoretic reason why the right hand side of Theorem 2is almost a modular form?See Kac’s beautiful article on modular invariance [10].3Connection with Symmetric Function TheoryTheorem 2has a very clean statement in terms of symmetric functions.For this,we recall the Hall-Littlewood polynomials (page 208of Macdonald [15]).Let the permutation w act on variables x 1,x 2,···by sending x i to x w (i ).There is also a coordinate-wise action of w on λ=(λ1,···,λn )and S λn is defined as the subgroup of S n stabilizing λin this action.Recall that m i(λ)is thenumber of parts ofλof size i.For a partitionλ=(λ1,···,λn)of length≤n,two definitions of the Hall-Littlewood polynomials are:Pλ(x1,···,x n;t)=[11−t ]w∈S nw(xλ11···xλn ni<jx i−tx jx i−x j)Atfirst glance it is not obvious that these are polynomials,but the denominators cancel out after the symmetrization.The Hall-Littlewood polynomials interpolate between the Schur functions (t=0)and the monomial symmetric functions(t=1).The statement of Theorem3uses the standard notation that n(λ)= i≥1(i−1)λi.Theorem3For k≥2,λ:λ1<k Pλ(1q2,···;1q n(λ)=∞r=1r=0,±k(mod2k+1)(1q r)Proof:The result follows from Theorem2and Macdonald’s principal specialization formula(page 337of Macdonald[15])which,when applied to the Hall-Littlewood polynomials,states thatPλ(1q2,···;1q n(λ)=1q)mi(λ) =1q)mi(λ)2Remarks1.Stembridge[16]also found some connections between the Rogers-Ramanujan identities andthe Hall-Littlewood symmetric functions,but the statement of Theorem3seems to be new.2.The Schur functions are well known to give rise to the irreducible polynomial representations ofthe general linear groups(see for instance Chapter3of Macdonald[15].Theorem3,together with the paper of Feigen and Frenkel[8],suggests that the Hall-Littlewood polynomials should also have a representation theoretic interpretation.4AcknowledgementsThis work is taken from the author’s Ph.D.thesis,done under the supervision of Persi Diaconis.His idea of studying the random partitionsλφled to this work.Ed Frenkel and Jim Lepowsky provided helpful pointers to the Rogers-Ramanujan literature.This research was done under the generous 3-year support of the National Defense Science and Engineering Graduate Fellowship(grant no. 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