Broadhurst s Dimensions

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1 Broadhurst s Dimesios May 10, 2010 Abstract Broadhurst cojectured a dimesio for the graded pieces of the Q vector space of multiple zeta values based o large scale experimetal computatios. I this talk, I will give a ew meaig to these dimesios by showig that his geeratig series is i fact the geeratig series for a vector space which is cojecturally isomorphic to the Q v.s. of multizeta values. 1 Backgroud Defiitio 1. MZV Defiitio 2. ζ(k) = ζ(k 1,..., k d ) := wt(k) = k i dp(k) = d. 1 > 2 > > d >0 Z := ζ(k) : wt(k) = Q Z d 1 k1 1 k d d := ζ(k) : dp(k) d Q Z, k i Z + k 1 2 Property 1. Z b l Z c m Z b+c l+m Defiitio 3. The vector space of eve period polyomials of weight, P polyomials P satisfyig Q[X], is the space of eve degree(p ) 4 P (X) + X 2 P ( 1 X ) = 0, P (X) + X 2 P (1 1 X ) + (X 1 1) 2 P ( 1 X ) = 0. Do says that this is the wrog defiitio of the period polyomial structure. Let (S w) to deote the coefficiet of the word or moomial w i a polyomial (or power series) S. Lemma 1. From the first relatio, oe may easily deduce that a weight period polyomial satisfies (P X 2i ) = (P X 2 2i ). 1

2 2 The Broadhurst Kreimer cojecture Broadhurst ad Kreimer cojectured the followig formula for Poicaré series for the algebra of multiple zeta values. Defiitio 4. D(X, Y ) O(X) := X3 1 X 2 X 12 S(X) := (1 X 4 )(1 X 6 ) = E(X) := X2 1 X 2 D 0 1 (X, Y ) := 1 OY + SY 2 SY 4 D(X, Y ) := (1 + EY ) D 0 (X, Y ). 12, eve dim(p )X Cojecture 2. (BK) D(X, Y ) =,d dim(z d /Z d 1 )X Y d. 3 Coefficiets i the B-K series ad multiple zeta values Remarks The coefficiet, E, correspods to followig the well-kow cojecture. Cojecture 3. Z := Z 0 Q[π 2 ]. Note that π 2 = kζ(2) where k Q ad that E is the Poicaré series for the for Q[X 2 ] Q[π 2 ]. 2. The coefficiet, O, correspods to aother well-kow cojecture. Cojecture 4. (Zagier) (Z 0 ) Q f i ; i 3, odd. Note that O is the Poicaré series for Q f i ad that if we igore the depth filtratio, i.e. set Y = 1, the D is the Poicaré series for the cojectured multizeta value algebra (from cojectures 3 ad 4) graded by weight. 3. To explai the appearace of S, we itroduce the followig algebras. Defiitio 5. Let N Z = Z / Z 2 >0 Q π 2 Q, which we kow by the work of Ecalle ad Raciet to be a Lie coalgebra. We defie FN Z to be the Lie coalgebra satisfyig all kow relatios betwee MZV s, ad we have that FN Z N Z. The dual of FN Z is a Lie algebra, deoted ds. The Lie algebra, ds ca be see as a subvector space of Q x, y ad it is graded by weight, which is the legth of the words. I Q x, y, the depth of a moomial is the umber of y i it. The elemets i ds are Q liear combiatios of words i varyig depth. We say that a elemet f ds has depth d if d = mi(depth(w); w has o-zero coefficiet i f). 2

3 The Lie algebra ds, is ot graded by depth but it possesses a descedig depth filtratio: ds 1 ds 2 ds 3. We kow at least that ds possesses at least oe elemet of depth 1 i each odd weight 3. Let {s = x 1 y+ terms of higher depth; 3, odd} be a set of depth oe geerators of ds. I [S], Scheps idetifies a Zagier period polyomial with a Poisso bracket of depth 1 elemets of ds by π : P F a ij (X i 1 X j 1 ) a ij {s i, s j }. i+j= i+j= Theorem 5. (Scheps) For all, Im(π ) ds 4. More precisely, the coefficiet of ay moomial of depth 3 or less i ay elemet i the image of π is 0. (This result was kow to Eichler ad Shimura i a differet cotext). 4 Costructio of a algebra satisfyig the B-K cojecture Defiitio 6. Let F be the o-commutative polyomial algebra over Q geerated by elemets of odd weight: F = Q s 3, s 5, s 7,.... The weight of a word i F is the sum of the idices. We therefore have a gradig of F by the weight: F = F. Do says that we should defie this as a tesor algebra, ot a cocateatio algebra. We defie a depth filtratio o F as follows. First, let F P be the image of the liear map P F i+j= =12, eve =0 a ij (X i 1 X j 1 ) i+j= a ij [f i, f j ]. We call the image of a period polyomial by this map a Lie period polyomial. Defiitio 7. Special depth filtratio V 1 := s i Q F V 2 := s i s j Q V 3 := s i s j s k Q V 4 := s i s j s k s l F P Q d 5, V d := ST : S V i, T V j, i + j = d Q F d := i d V i Q. Do says that we should ot use uios of spaces. We caot geerate a space with a space. Nobody does that. Sice F is graded by weight, this defiitio gives us a filtratio o the weight graded pieces: F = F 1 F 2 F /3. Notice that the degree ad the depth of the polyomials i F correspod up to a factor of a (or may) period polyomial(s). This defiitio is ispired by Broadhurst ad Kreimer s series, because the deomiator is tellig us that we should take F P out of depth 2 ad put it i depth 4. 3

4 Theorem 6. (CGS) If cojecture 7 holds, the dimesios of the graded parts associated to the depth filtratio of F are give by the Broadhurst-Kreimer series, D 0 : Cojecture 7. dim(f d /F d+1 ) = ( D 0 x y d). F P V 1 Q V 1 F P Q = 0. More precisely, we wat to show that for ay give odd, ad ay set of period polymomials, each of weight j <, {P j }, the the followig polyomials are always liearly idepedet, {X j 1 P j (Y ) + Y j 1 P j (X) : j < }. Proof. From the expressio for D 0, we have that ( )( ) a + b + c a + b (D 0 y d ) = S a+b O c, so that ( (D 0 x y d ) = 4a+2b+c=d 4a+2b+c=d ( )( ) a + b + c a + b ( 1) b S a+b O c x ). Now, all we have to do is show that this is the same as the dimesio of F d /F d+1. To show that this is ideed the geeratig series, we first idetify subspaces of F d,k F d, which are geerated by polyomials of degree k. This provides a bi-gradig o F d by the degree ad the weight. Remarks 2. Whe d < k, the F d,k = F d+1,k. This is because each V i is geerated by moomials of degree i. So we have: F d.k i k V i Q i d+1 V i Q = F d+1,k Remarks 3. Whe d k is odd, k < d, the F d,k = F d+1,k. F d,k. By defiitio, the degree ad the depth oly vary by factors of F P which is of degree 2 ad of maximal depth 4. Therefore the appearace of a factor of a Lie period polyomial i a elemet of F icreases the depth of that elemet by 2. From remarks 1 ad 2, we have the expressio, F/F d d+1 = F,d/F d d+1,d F,d 2/F d d+1,d 2 F,d 2 d/4 d d+1 /F,d 2 d/4. (1) The key to the proof is to write (for a positive): F d,d 2a (j 0,...,j a) ( (F 1 ) j0 F P (F 1 ) j1 F P (F 1 ) j ) a = N (E J i ), J := {(j 0,..., j a ); Σj i = d 4a, j i 0} = {J 1,..., J N }. (2) This is a isomorphism because this vector space cotais all elemets with degree d 2a ad special depth d, sice there are a factors of F P i there. I order to cout the dimesio of F,d 2a d, the we eed to determie the dimesio of the itersectio of the vector spaces i the sum (2). The Möbius iversio formula for the dimesio of vector spaces is aalogous to the iclusio/exclusio priciple for sets: ( dim(f,d 2a) d N ) = E J i i=0 We eed to establish the followig lemmas. = T [1,N], T 1 ( 1) T 1 ( i T i=1 ) E J i. (3) 4

5 Lemma 8. This is true because F P (F 1 ) 2. It implies that Cojecture 7 tells us that (F 1 ) 2 F P F P (F 1 ) 2 = F P F P. i T E Ji = F a+b P (F 1 ) c. E 1,... E 0,... = (F 1 F P ) (F P ) = 0 because the first factor of F 1 o the left had side crosses the first factor of F P o the right had side. I the geeral case, the dimesio cout reduces to coutig o-crossig partitios of d 4a. We proved that the terms o the right had side of equatio (3) regroup to give the dimesio, ( )( ) a + b + c a + b 1 ( 1) b ( F a+b P (F 1 ) c). The Sice the geeratig series for F P is S ad the geeratig series for F 1 is O, by equatio (3) we have ( )( ) a + b + c a + b 1 (S dim(f,d 2a) d = ( 1) b a+b O c x y d ). dim(f d,d 2a/F d+1,d 2a ) = ( )( ) a + b + c a + b (S ( 1) b a+b O c x y d ). By takig the sum ow over a from equatio (1), we have that ( )( ) a + b + c a + b (S dim(f/f d d+1 ) = ( 1) b a+b O c x y d ). 2b+c+4a=d a list of positive itegers: Refereces [IKZ] K. Ihara, M. Kaeko, D. Zagier, Derivatio ad double shuffle relatios for muliple zeta values, Compositio Math. 142 (2006), pp [R] G. Raciet, Séries géératrices o-commutatives de polyzêtas et associateurs de Drifel d, Thèse de doctorat, Uiversité de Picardie-Jules Vere (2000) [S] L. Scheps, O the Poisso bracket o the free Lie algebra i two geerators, J. Lie Theory, Vol. 16 (2006), pp [Z] D. Zagier, Quelques coséqueces surpreates de la cohomologie de SL 2 (Z), Leços de Mathématiques d Aujourd hui, E. Charpetier ad N. Nikolski, eds., Cassii, Paris (2000), pp