REMARKS ON THE PAPER SKEW PIERI RULES FOR HALL LITTLEWOOD FUNCTIONS BY KONVALINKA AND LAUVE

Transcription

1 REMARKS ON THE PAPER SKEW PIERI RULES FOR HALL LITTLEWOOD FUNCTIONS BY KONVALINKA AND LAUVE S OLE WARNAAR Abstract In a recent paper Konvalinka and Lauve proved several skew Pieri rules for Hall Littlewood polynomials In this note we show that q-analogues of these rules are encoded in a q-binomial theorem for Macdonald polynomials due to Lascoux and the author The Konvalinka Lauve formulas and their q-analogues We refer the reader to [5] for definitions concerning Hall Littlewood and Macdonald polynomials Let P / = P / (X; t and Q / = Q / (X; t be the skew Hall Littlewood polynomials, e r = P ( r the rth elementary symmetric function, h r the rth complete symmetric function and q r = Q (r Then the ordinary Pieri formulas for Hall Littlewood polynomials are given by [5] (a P e r = vs / (tp (b P q r = hs / (tp, where the sums on the right are over partitions such that = + r The Pieri coefficient vs / (t is given by [5, p 25, (32] (2 vs / (t = [ i ] i+ i i, i t so that vs / (t is zero unless with a vertical strip Similarly, hs / (t vanishes unless with a horizontal strip, in which case [5, p 28, (30] ( (3 hs / (t = t i i+ i = i + i+ = i+ To express the skew Pieri formulas, Konvalinka and Lauve [9] (see also [8] introduced a third Pieri coefficient (4 sk / (t := t [ n(/ i ] i+ i i, i t where n(/ := ( i Note that sk/ (t = 0 if i i Mathematics Subject Classification 33D52,05E05 Work supported by the Australian Research Council

2 2 S OLE WARNAAR It seems Konvalinka and Lauve have been unaware that the above function has appeared in the literature before Indeed, in exactly the above form and denoted as g (t, it was used by Kirillov to prove the Pieri rule [7, Lemma 4] (5 P h r = sk / (tp Moreover, sk / (t arose in [20, Equation (43] as a formula for the modified Hall Littlewood polynomial Q / ( = Q /(, t, t 2, a result first stated in [2, Theorem 3], albeit in the not-so-easily-recognisable form l( Q / ( = t n(/ t i i+ for, (t; t i= i i+ 0 otherwise In a more general form pertaining to Macdonald polynomials it also appeared in [8, p 73, Remark 2] and [9, Proposition 32], see (8 below Prior to the abovementioned papers sk / (t appeared in the theory of abelian p-groups: sk / (t = t n( n( α (; t, where α (; p is the number of subgroups of type in a finite abelian p-group of type, [2 4, 2] Theorem (Konvalinka Lauve [9, Theorems 2 4] For partitions, (6a P / e r =,η ( η vs / (t sk /η (tp /η (6b (6c P / h r =,η ( η sk / (t vs /η (tp /η P / q r =,η,ω( ω t ω η hs / (t vs /ω (t sk ω/η (tp /η, where each of the multiple sums is subject to the restriction + η = + + r For = 0 the first and third skew Pieri formulas reduce to (a and (b respectively, whereas the second formula simplifies to (5 (see also [9, Theorem ] Theorem for t = 0 gives the skew Pieri rules for Schur functions due to Assaf and McNamara [] who, more generally, conjectured a skew Littlewood Richardson rule The identities (6a and (6b were first conjectured by Konvalinka in [8] The subsequent proof of the theorem by Konvalinka and Lauve combines Hopf algebraic techniques in the spirit of the proof of the Assaf McNamara conjecture [0] with intricate manipulations involving t-binomial coefficients The aim of this note is to point out that all of the skew Pieri formulas (6a (6c are implied by a generalised q-binomial theorem for Macdonald polynomials and, consequently, have simple q-analogues From here on let P / = P / (X; q, t and Q / = Q / (X; q, t denote skew Macdonald polynomials Let f be an arbitrary symmetric function Adopting plethystic or -ring notation, see eg, [5, ], we define f ( (a b/( t in terms of the power sums with positive index r as p r ( a b t = ar b r t r

3 SKEW PIERI RULES 3 In other words, p r ( (a b/( t = a r ɛ b/a,t (p r with ɛ u,r Macdonald s evaluation homomorphism [5, p 338, (66] Equivalently, in terms of complete symmetric functions, ( a b h r = [z r ] (bz; t t (az; t We now define the following five Pieri coefficients for Macdonald polynomials: (7a (7b (7c (7d (7e ( q vs / (q, t := ψ / (q, t = ( Q / t hs / (q; t := ϕ / (q, t = Q / ( ( q sk / (q, t := Q / t ( q/t ŝk / (q, t := Q / t ks / (q, t := Q / (, where ψ / (q, t and ϕ /(q, t is notation used by Macdonald, and where the in Q / ( is a plethystic, ie, applied to the power sum p r of positive index r it gives the number The Pieri coefficients vs / (q, t and hs / (q, t have nice factorised forms generalising (2 and (3, see [6, pp ] So does ŝk / (q, t [8, p 73, Remark 2], [9, Proposition 32]: l( t n( n( (qt j i ; q i j (qt j i ; q i j (8 ŝk / (q, t = (qt j i ; q i,j= i j (qt j i for, ; q i j 0 otherwise, where (a; q k := (a; q /(aq k ; q for all k Z We leave it to the reader to verify that the above right-hand side for q = 0 reduces to the right-hand side of (4 The remaining two Pieri coefficients do not factor into binomials For example sk (2,/(,0 (q, t = q q2 + t + qt q 2 t q 2 t ks (2,/(,0 (q, t = ( t( + q t + qt t2 qt 2 ( q( q 2 t Of course, sk / (0, t = sk / (t so it does factorise in the classical limit This is however not the case for ks / (0, t, and ks (2,/(,0 (0, t = ( t( t t 2 Let g r = g r (X; q, t = Q (r (X; q, t, so that g r (X; 0, t = q r (X; t Then the following q-analogue of Theorem holds

4 4 S OLE WARNAAR Theorem 2 For partitions, (9a P / e r =,η ( η vs / (q, t sk /η (q, tp /η (9b P / h r =,η ( η sk / (q, t vs /η (q, tp /η (9c (9d P / g r =,η hs / (q, t ks /η (q, tp /η =,η,ω( ω t ω η hs / (q, t vs /ω (q, t ŝk ω/η(q, tp /η, where each of the multiple sums is subject to the restriction + η = + + r 2 The q-binomial theorem for Macdonald polynomials In [4, Equation (2] Lascoux and the author proved the following q-binomial theorem for Macdonald polynomials: ( (bx; q (2 Q / P / (X = Q / P / (X t (ax; q t For = = 0 and (a, b (, a this is the well-known Kaneko Macdonald q- binomial theorem [6, 6] (22 t n( (a c P (X = where we have used that [5, p 338, (67] ( a Q = t n( (a t c (ax; q (x; q, Here (a = (a; q, t := i (at i ; q i and c = c (q, t is the generalised hook polynomial c = ( s q a(s+ t l(s with a(s and l(s the arm-length and leg-length of the square s To show that (2 encodes the skew Pieri formulas (9a (9d we first consider the = 0 case (23 Q / P (X = P (X (bx; q t (ax; q ( If we multiply this by Q / (b a/( t and sum over using (23 with (,, a, b (,, b, a we obtain ( b a Q / Q / P (X = P (X t t, This implies the orthogonality relation (implicit in [7] and given in its more general nonsymmetric form in [3, Equation (65] ( b a (24 Q / Q / = δ t t

5 SKEW PIERI RULES 5 Thanks to (24, identity (2 is equivalent to ( b a Q /η Q / P /η (X = P / (X (ax; q t t (bx; q,η There are now three special cases to consider First, if b = aq then P / (X ( ax = ( q ( q a + η Q / Q /η P /η (X t t,η Equating coefficients of ( a r and using definition (7 yields (9a Next, if a = bq P / (X bx = ( q ( q b + η Q / Q /η P /η (X t t,η Equating coefficients of b r and again using (7 yields (9b Finally, if a = bt P / (X (btx; q (bx; q =,η b + η Q / (Q /η ( P /η (X, Equating coefficients of b r and using (7 gives (9c To show that (9c and (9d are equivalent, we recall Rains q-pfaff Saalschütz summation for Macdonald polynomials [7, Corollary 49]: (25 (a ( a b Q / (c t Q / ( b c t = (a ( (b a c Q /, (b (c t which for c = a is (24 Setting b = a/q and c = a/t and using (7 yields ks / (q, t = (t/q (a/q (a/t ( (a vs / (q, t (a (a/q (a/t ŝk /(q, t Taking the a limit this further simplifies to ks / (q, t = ( t vs / (q, t ŝk /(q, t, which proves the equality between (9c and (9d To conclude let us mention that all other identities of [9] admit simple q-analogues For example, if we take (25 and specialise b = a/q and c = at then (a ( qt ( vs / (q, tq / = (a (a/q q hs / (q, t (at t (a/q (at Letting a this reduces to ( qt ( t vs / (q, tq / = hs / (q, t t For q = 0 this is [9, Lemma 5] ( t vs / (t sk / (t = hs / (t Similarly, according to [3, Equation (623] (26 t n( (a c f (q, t = Q / ( a t

6 6 S OLE WARNAAR For a = q = 0 this is [7, Corollary 42], [9, Corollary 6] t n( f(t = sk / (t Finally, to obtain a q-analogue of [9, Theorem 7] we have to work a little harder First note that P (Xe m (X sk / (q, tp (Xe m (X (27 r=0 h r (X = = =m vs / (q, t sk / (q, tp (X To compute this in a different way, observe that if we set a = q in (22 then t n( (q c Using this as well as e m = P ( m we get P (Xe m (X P (X = x = h r (X r=0 r=0 h r (X = η t n(η (q η c η P (XP η (XP (m (X By a double use of P P = fp this leads to P (Xe m (X h r (X = t n(η (q η c P (XP η (XP ( m (X r=0 η η =,η =,,η t n(η (q η c η t n(η (q η c η f η,( m (q, tp (XP (X f η,( m (q, tf (q, tp (X (28 =, sk /(m (q, tf (q, tp (X, where the final equality follows from the a = q case of (26 Equating coefficients of P (X in (27 and (28 yields vs / (q, t sk / (q, t = sk /( m (q, tf(q, t =m By (4, [ ] [ ] sk /(m (0, t = sk /(m (t = t n(/(m = t n( (m 2 m t m so that for q = 0 we obtain [9, Theorem 7] vs / (t sk / (t = [ ] t n( (m 2 f (t m t =m Acknowledgements I thank Matjaž Konvalinka and Aaron Lauve for helpful discussions t,

7 SKEW PIERI RULES 7 References S H Assaf and P R W McNamara, A Pieri rule for skew shapes, J Combin Theory Ser A 8 (20, L M Butler, Subgroup Lattices and Symmetric Functions, Mem Amer Math Soc 2 (994, no S Delsarte, Fonctions de Möbius sur les groupes abelian finis, Ann Math 49 (948, P E Djubjuk, On the number of subgroups of a finite abelian group, Izv Akad Nauk USSR, Ser Mat 2 (948, J Haglund, The q, t-catalan Numbers and the Space of Diagonal Harmonics, Univ Lecture Ser, vol 4, Amer Math Soc, Providence, RI, J Kaneko, q-selberg integrals and Macdonald polynomials, Ann Sci Ecole Norm Sup (4 29 (996, A N Kirillov, New combinatorial formula for modified Hall Littlewood polynomials, in q- Series from a Contemporary Perspective, pp , Contemp Math Vol 254, AMS, Providance, RI, 2000, 2 8 M Konvalinka, Skew quantum Murnaghan Nakayama rule, J Algebraic Combin 35 (202, , 9 M Konvalinka and A Lauve, Skew Pieri rules for Hall Littlewood functions, J Algebraic Combin, to appear,,, 2, 2, 2 0 T Lam, A Lauve and F Sottile, Skew Littlewood Richardson rules from Hopf algebras, Int Math Res Not IMRN 20, A Lascoux, Symmetric Functions and Combinatorial Operators on Polynomials, CBMS Regional Conference Series in Math Vol 99, AMS, Providance, RI, A Lascoux, Adding to the argument of a Hall Littlewood polynomial, Sém Lothar Combin 54 (2005/07 Art B54n, 7 pp 3 A Lascoux, E M Rains and S O Warnaar, Nonsymmetric interpolation Macdonald polynomials and gl n basic hypergeometric series, Transform Groups 4 (2009, , 2 4 A Lascoux and S O Warnaar, Branching Rules for symmetric Macdonald polynomials and sl n basic hypergeometric series, Adv in Applied Math 46 (20, I G Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Clarendon Press, Oxford, 995,,,, 2 6 I G Macdonald, Hypergeometric series II, unpublished manuscript, 2 7 E M Rains, BC n-symmetric polynomials, Transform Groups 0 (2005, , 2 8 E M Rains, BC n-symmetric abelian functions, Duke Math J 35 (2006, 99 80, 9 S O Warnaar, q-selberg integrals and Macdonald polynomials, Ramanujan J 0 (2005, , 20 S O Warnaar and W Zudilin, Dedekind s η-function and Rogers Ramanujan identities, Bull Lond Math Soc 44 (202, 2 Y Yeh, On prime power Abelian groups, Bull Amer Math Soc 54 (948, School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia