THE TUTTE POLYNOMIAL OF A FINITE PROJECTIVE SPACE

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1 THE TUTTE POLYNOMIAL OF A FINITE PROJECTIVE SPACE MICHAEL BARANY AND VICTOR REINER Abstract We comute a -exoetial geeratig fuctio collatig the Tutte olyomials for the family of matroids comig from fiite rojective saces 1 The geeratig fuctio Fix a rime ower, ad cosider the arragemet A(, ) cosistig of all ] : 1 1 ossible hyerlaes i F Alteratively, these hyerlaes have ormal vectors give by the colums of a ] matrix, cotaiig oe vector from each lie i F The oit of these otes is to comute a comact geeratig fuctio for the Tutte olyomials T A(,) (x, y); a exlicit formula for each T A(,) (x, y), equivalet to (3) below was comuted by Mhako 5] The geeratig fuctio is -exoetial, ad uses some of these basic hyergeometric otatios: (x; ) : (1 x)(1 x)(1 2 x) (1 1 x) (x; ) : (1 x)(1 x)(1 2 x) ] : ]! : ] 1] 2] 1] (; ) ]! : (; ) l (; ) l!! Theorem 1 T A(,) (x, y) u (y 1) (u; ) (; ) ((x 1)(y 1)u; ) y k] (; ) k Date: May 2005 Key words ad hrases Tutte olyomial, fiite rojective sace, fiite field method This work was isired by the C code for comutig Tutte olyomials roduced by the first author as their Srig 2005 UROP roject, sosored by the Uiv of Miesota Udergraduate Research Oortuities Program, ad available at wwwmathumedu/ reier Secod author suorted by NSF grat DMS

2 2 MICHAEL BARANY AND VICTOR REINER Proof We emloy the fiite field method exosed i 1, 3] Here oe comutes istead the equivalet coboudary olyomial χ A(,) (q, t) : x F q t h(x) where q r is some ower of, so that F q is a field extesio of F, ad where h(x) is the umber of hyerlaes i A(, ) o which the vector x F q lies This χ A (q, t) will be a olyomial i q ad t, related to the Tutte olyomial as follows 1 : T A (x, y) 1 (y 1) χ rak(a) A ((x 1)(y 1), y) (1) To comute χ A(,) (q, t), we take advatage of the F -vector sace isomorhism F q F r to rereset a vector x (x 1,, x ) as a r matrix over F, whose i th colum reresets x i If this matrix has rak l, the x reresets a vector that will lie o exactly h(x) hyerlaes i A(, ) Cosequetly, if we ca cout the umber of r matrices over F havig rak l, we ca assemble the coefficiets of χ A(,) (q, t) It turs out that there are l 1 ( r i 1) i0 l 1 (q i 1) i0 q l (q 1 ; ) l such matrices, usig the fact that GL r (F) GL (F) acts trasitively o them, ad calculatig the stabilizer subgrou of a tyical rak l matrix Cosequetly, χ A(,) (q, t) l0 (2) q l (q 1 ; ) l t (3) 1 We are lyig slightly here: the fiite field method exosed i 1, 3] assumes a arragemet of hyerlaes with ormal vectors i Z d, ad cosiders a coutig roblem for the reduced arragemet i F d q for various rimes owers q However, it alies equally well to a arragemet of hyerlaes i F d for a rime ower, which oe the cosiders as a arragemet i F d q for various owers q r ; this is the cotext of Crao ad Rota s critical roblem 3, 16]

3 THE TUTTE POLYNOMIAL OF A FINITE PROJECTIVE SPACE 3 This assembles icely ito a -exoetial geeratig fuctio u χ A(,) (q, t) ( ) q (; ) l (q 1 u ; ) l t l0 (; ) q l (q 1 ; ) l u l t u l (; ) l (; ) l l0 (q 1 ; ) l (qu) l t k] (; ) l (; ) k l 0 The first sum o the last lie ca be evaluated as a ifiite roduct by the -biomial theorem (a; ) l x l (ax; ) (; ) l (x; ) l 0 Hece takig a q 1 ad x qu, oe obtais u χ A(,) (q, t) (u; ) (; ) (qu; ) t k] (; ) k (4) Accordig to (1), we should ow substitute q (x 1)(y 1) ad t y After otig that rak(a(, )), the theorem follows 2 Kow secializatios Here are two well-kow secializatios of the foregoig calculatios 21 The characteristic olyomial Settig t 0 i (3) (or equivaletly, settig l i (2)) yields the umber of vectors i F q that lie o oe of the hyerlaes i A(, ), which is equivalet (u to rescalig) to the characteristic olyomial of the matroid of A(, ): q (q 1 ; ) (q 1)(q )(q 2 ) (q 1 ) 22 Dual Hammig ad Hammig codes The theorem ca be used to derive the weight eumerator A(z) for the dual Hammig code, whose code vectors cosist of the -dimesioal row-sace i F ] for the ] matrix that reresets the matroid A(, ) Greee 4] showed that the weight eumerator is related to the Tutte olyomial by ( 1 + ( 1)z A(z) (1 z) z ] T A(,), 1 ) 1 z z He comutes 4, Examle 34] that the dual Hammig code has the extremely simle weight eumerator A(z) 1 + ( 1)z 1 (5)

4 4 MICHAEL BARANY AND VICTOR REINER Ideed this follows from the theorem with a little algebra, otig that the secializatio x 1+( 1)z 1 z ad y 1 z leads to the relatio (x 1)(y 1), ad usig the fact that (u; ) 1 u (u; ) Oe ca, of course, also deduce from the theorem the weight eumerator for the Hammig code itself, rather tha its dual But this also follows from (5) via the MacWilliams idetity (see 4]) 3 Alterate aroach: -coes Lastly, we metio a alterate aroach to the derivatio of Theorem 1 I 2], the authors derive a ice formula exressig the Tutte olyomial T M (x, y) for the -coe 2 M of a matroid M of rak r rereseted iside a fiite rojective sace P r F, i terms of T M (x, y) Phrased istead i terms of the coboudary olyomials, their formula reads χ M (q, t) tχ M (q, t ) + r (q 1)χ M ( q, t ) (6) Oe ca costruct the tower of fiite rojective geometries P F by iteratig this -coe costructio, begiig with the seed geometry M 0 P 1 F of rak 0 The the -exoetial geeratig fuctio u F (q, t, u) : χ A(,) (q, t) (; ) obeys the followig recurrece derived from (6): ( ) q F (q, t, u) tuf (q, t, u) u(q 1)F, t, u + F (q, t, u) (7) O the face of it, this recurrece looks hard to solve However, with the hidsight of formula (4) which oe hoes to derive for F (q, t, u), it is better to rehrase this recurrece i terms of the geeratig fuctio ˆF (q, t, u) : (qu; ) (u; ) F (q, t, u), which we exect to (miraculously!) be ideedet of q The recurrece (7) becomes ˆF (q, t, u) tu ˆF (q, t, u) 1 ( ) ] u(q 1) 1 u ˆF q, t, u + (1 qu) ˆF (q, t, u) (8) 2 Here is a defiitio of the -coe costructio M, startig with a matroid M rereseted by oits i P r F First embed P r F i P r+1 F The choose a aex oit a i P r+1 F P r F The let M be the uio of all lies saed by a together with oits of M

5 THE TUTTE POLYNOMIAL OF A FINITE PROJECTIVE SPACE 5 Oe ca use this last recurrece to rove that the coefficiet of u i ˆF (q, t, u) is ideedet of q by iductio o With this kowledge i had, the recurrece (8) the greatly simlifies to ˆF (t, u) tu ˆF (t, u) ˆF (t, u) (9) This is easily solved (eg by writig dow the recurrece it gives for the coefficiet of o both sides), yieldig u (;) i agreemet with (4) ˆF (q, t, u) t k] (; ) k, Ackowledgemets The authors thak Joe Boi for oitig them to the referece 5] ad suggestig that they ursue a aroach as i Sectio 3 They also thak Deis Stato for useful discussios regardig the recurrece (7) Refereces 1] F Ardila, Comutig the Tutte olyomial of a hyerlae arragemet ArXiv rerit, 2004, mathco/ ] JE Boi ad H Qi, Tutte olyomials of q-coes Discrete Math 232 (2001), ] HH Crao ad G-C Rota, O the foudatios of combiatorial theory: Combiatorial geometries Prelimiary editio MIT Press, Cambridge, Mass-Lodo, ] C Greee, Weight eumeratio ad the geometry of liear codes Studies i Al Math 55 (1976), ] EG Mhako, Tutte olyomials of erfect matroid desigs Combi Probab Comut 9 (2000), School of Mathematics, Uiversity of Miesota, Mieaolis, MN 55455, USA address: bara0051@umedu, reier@mathumedu