Factorisations of the Garside element in the dual braid monoids

Factorisations of the Garside element in the dual braid monoids Vivien Ripoll École Normale Supérieure Département de Mathématiques et Applications 30 June 2010 Journées Garside Caen

1 Dual braid monoids and noncrossing partition lattices The dual braid monoid Factorisations in the noncrossing partition lattice 2 Factorisations from the geometry of the discriminant The Lyashko-Looijenga covering Factorisations as fibers of LL Combinatorics of the submaximal factorisations

Classical setting: Artin-Tits monoids W finite real reflection group, S system of fundamental reflections.

Classical setting: Artin-Tits monoids W finite real reflection group, S system of fundamental reflections. Artin-Tits monoid of (W, S) : A + (W, S) := S s, t S, sts }{{.. }. = }{{} tst... m s,t m s,t Mon. Alternative presentation [Tits] : W w.w = ww whenever l S (w) + l S (w ) = l S (ww ) Mon. Garside monoid; Garside element: (copy of) w 0 (the longest element); simple elements: (copy of) W = {w W w S w 0 } (lattice for S ),

Dual braid monoid Basic idea: replace S with R := {all reflections in W }. new definition of length (l R ) and of partial order ( R ).

Dual braid monoid Basic idea: replace S with R := {all reflections in W }. new definition of length (l R ) and of partial order ( R ). Definition (Dual braid monoid of W ) M(W, c) is the monoid with presentation [1, c] w.w = ww if l R (w) + l R (w ) = l R (ww ). Garside monoid; Garside element: (copy of) c (a Coxeter element); simple elements: (copy of) [1, c] = {w W w R c} (lattice for R );

Complex reflection groups V : complex vector space, of dim. n. Definition A (finite) complex reflection group is a finite subgroup of GL(V ) generated by complex reflections. A complex reflection is an element s GL(V ) of finite order, s.t. Ker(s Id V ) is a hyperplane: s B matrix Diag(ζ, 1,..., 1), with ζ root of unity.

Invariant theory W a complex reflection group. W acts on S(V ) (polynomial algebra C[v 1,..., v n ]).

Invariant theory W a complex reflection group. W acts on S(V ) (polynomial algebra C[v 1,..., v n ]). Theorem (Chevalley-Shephard-Todd) There exist fundamental invariant polynomials f 1,..., f n (homogeneous), s.t. S(V ) W = C[f 1,..., f n ]. Their degrees d 1 d n do not depend on the choice of f 1,..., f n (invariant degrees of W ).

The noncrossing partition lattice Suppose W irreducible, well-generated. Let c be a fixed Coxeter element (i.e. e 2iπ/h -regular, where h = d n ).

The noncrossing partition lattice Suppose W irreducible, well-generated. Let c be a fixed Coxeter element (i.e. e 2iπ/h -regular, where h = d n ). Definition (Noncrossing partition lattice of type W ) NCP W (c) := {w W w R c} (the structure does not depend on the choice of c.)

Multichains in NCP W Chapoton s formula The number of multichains w 1 R... R w N R c in NCP W is : Z W (N + 1) = n i=1 d i + Nh d i.

Multichains in NCP W Chapoton s formula The number of multichains w 1 R... R w N R c in NCP W is : Z W (N + 1) = n i=1 d i + Nh d i. Called Fuss-Catalan numbers of type W : Cat (N) (W ).

Multichains in NCP W Chapoton s formula The number of multichains w 1 R... R w N R c in NCP W is : Z W (N + 1) = n i=1 d i + Nh d i. Called Fuss-Catalan numbers of type W : Cat (N) (W ). Proof (Athanasiadis, Reiner, Bessis): case-by-case using the classification... even for N = 1 (formula for NCP W ).

Block factorisations of c (will appear naturally in the geometry of B(W ))

Block factorisations of c (will appear naturally in the geometry of B(W )) Definition (w 1,..., w p ) is a block factorisation of c if :

Block factorisations of c (will appear naturally in the geometry of B(W )) Definition (w 1,..., w p ) is a block factorisation of c if : w 1... w p = c ;

Block factorisations of c (will appear naturally in the geometry of B(W )) Definition (w 1,..., w p ) is a block factorisation of c if : w 1... w p = c ; w 1,..., w p W {1};

Block factorisations of c (will appear naturally in the geometry of B(W )) Definition (w 1,..., w p ) is a block factorisation of c if : w 1... w p = c ; w 1,..., w p W {1}; l R (w 1 ) + + l R (w p ) = l R (c) = n (i.e. w 1 R w 1 w 2 R... R w 1... w p 1 R c). FACT p (c) := {factorisations in p blocks} determines a partition of n, and even a composition (ordered partition) of n.

Factorisations vs multichains Combinatorics of factorisations: similar to multichains. But factors must be non-trivial ( strict chains).

Factorisations vs multichains Combinatorics of factorisations: similar to multichains. But factors must be non-trivial ( strict chains). Conversion formulas n ( ) Cat (N) N + 1 (W ) = FACT k (c) k k=1 p ( ) p FACT p (c) = p Z W (0) = ( 1) p k Cat (k) (W ) k ( : P P(X + 1) P(X).) k=1

Discriminant of W A := {reflection hyperplanes of W }. For H A: α H : linear form with kernel H; e H : order of the parabolic subgroup W H. Definition Discriminant of W : W := H A α e H H.

Discriminant of W A := {reflection hyperplanes of W }. For H A: α H : linear form with kernel H; e H : order of the parabolic subgroup W H. Definition Discriminant of W : W := H A α e H H. W C[V ] W = C[f 1,..., f n ] W is the equation of the hypersurface H, quotient of H A H, in W \V Cn.

Discriminant of a well-generated group Suppose W acts irreducibly on V (of dim. n), and is well-generated (i.e. can be generated by n reflections).

Discriminant of a well-generated group Suppose W acts irreducibly on V (of dim. n), and is well-generated (i.e. can be generated by n reflections). Proposition If W is well-generated, the discriminant W is monic of degree n in f n. The fundamental invariants f 1,..., f n can be chosen s.t.: W = f n n + a 2 f n 2 n + a 3 f n 3 n + + a n 1 f n + a n, with a i C[f 1,..., f n 1 ] (homogeneous polynomial of degree ih).

Lyashko-Looijenga morphism of type W Definition (Lyashko-Looijenga morphism) LL : C n 1 C n 1 (f 1,..., f n 1 ) (a 2,..., a n ) It is an algebraic morphism, which is quasi-homogeneous for the weights deg(f j ) = d j, deg(a i ) = ih. Define Y := Spec C[f 1,..., f n 1 ]. W \V Y C. LL : Y E n = {configurations of n points in C} y {roots, with multiplicities, of W (y, f n ) in f n }

Lyashko-Looijenga covering E reg n := {configurations of n distincts points} E n K := LL 1 (E n En reg ) = {y Y W (y, f n ) has multiple roots in f n } = {y Y D LL (y) = 0},

Lyashko-Looijenga covering E reg n := {configurations of n distincts points} E n K := LL 1 (E n En reg ) = {y Y W (y, f n ) has multiple roots in f n } = {y Y D LL (y) = 0}, D LL := Disc( W (y, f n ); f n ) = Disc(fn n + a 2 fn n 2 + + a n ; f n ). with Theorem (Looijenga, Lyashko, Bessis) The extension C[a 2,..., a n ] C[f 1,..., f n 1 ] is free, with rank n!h n / W. LL is a finite morphism. its restriction Y K En reg is an unramified covering of degree n!h n / W.

Factorisations arising from topology Hypersurface H W \V Y C. (y, x) H x LL(y)

Factorisations arising from topology Hypersurface H W \V Y C. (y, x) H x LL(y) Topological constructions by Bessis (tunnels) a map H W (y, x) c y,x

Factorisations arising from topology Hypersurface H W \V Y C. (y, x) H x LL(y) Topological constructions by Bessis (tunnels) a map H W (y, x) c y,x s.t., if (x 1,..., x p ) is the ordered support of LL(y) (for the lex. order on C R 2 ), then: (c y,x1,..., c y,xp ) FACT p (c). Notation : fact(y) := (c y,x1,..., c y,xp ).

Stratification of W \V and parabolic Coxeter elements Stratification of V with the flats (intersection lattice) : L := { H B H B A}.

Stratification of W \V and parabolic Coxeter elements Stratification of V with the flats (intersection lattice) : L := { H B H B A}. Quotient by W stratification L = W \L = (W L) L L.

Stratification of W \V and parabolic Coxeter elements Stratification of V with the flats (intersection lattice) : L := { H B H B A}. Quotient by W stratification L = W \L = (W L) L L. For Λ L, Λ 0 := Λ minus its strata strictly included. Bijection [Steinberg] : L {parabolic subgroups of W } =: PSG(W ). natural bijections : L PSG(W )/conj.

Factorisations and LL Proposition Fix y Y. For all x LL(y), c y,x is a parabolic Coxeter element of W. Its length is the multiplicity of x in LL(y); its conjugacy class corresponds to the unique stratum Λ in L s.t. (y, x) Λ 0. Block factorisations composition of n ω E n composition of n (multiplicities in the lex. order) Prop. compatibility of fact(y) and LL(y) (same comp.) Theorem (Bessis) LL fact The map Y E n FACT(c) is injective, and its image is the set of compatible pairs.

Reduced decompositions of c Red R (c) = FACT µ (c) for µ = (1,..., 1)

Reduced decompositions of c Red R (c) = FACT µ (c) for µ = (1,..., 1) = LL 1 (ω) for ω En reg

Reduced decompositions of c Red R (c) = FACT µ (c) for µ = (1,..., 1) = LL 1 (ω) for ω En reg = deg(ll) = n!hn W

Reduced decompositions of c Red R (c) = FACT µ (c) for µ = (1,..., 1) = LL 1 (ω) for ω En reg = deg(ll) = n!hn W Can we compute in the same way FACT n 1 (c) = FACT 2 1 1 n 2(c)?

Irreducible components of K Ramified part of LL : K E n E reg n. K = {y Y D LL (y) = 0}.

Irreducible components of K Ramified part of LL : K E n E reg n. K = {y Y D LL (y) = 0}. L 2 := {strata of L with codimension 2}

Irreducible components of K Ramified part of LL : K E n E reg n. K = {y Y D LL (y) = 0}. L 2 := {strata of L with codimension 2} ϕ : W \V Y C Y v = (y, x) y

Irreducible components of K Ramified part of LL : K E n E reg n. K = {y Y D LL (y) = 0}. L 2 := {strata of L with codimension 2} ϕ : W \V Y C Y v = (y, x) y Proposition The irreducible components of K are the ϕ(λ), for Λ L 2.

Restriction of LL to a component of K Write D LL = Λ L 2 D r Λ Λ, with D Λ irreducibles in C[f 1,..., f n 1 ] s.t. ϕ(λ) = {D Λ = 0}.

Restriction of LL to a component of K Write D LL = Λ L 2 D r Λ Λ, with D Λ irreducibles in C[f 1,..., f n 1 ] s.t. ϕ(λ) = {D Λ = 0}. Define the restriction LL Λ : ϕ(λ) E n E reg n.

Restriction of LL to a component of K Write D LL = Λ L 2 D r Λ Λ, with D Λ irreducibles in C[f 1,..., f n 1 ] s.t. ϕ(λ) = {D Λ = 0}. Define the restriction LL Λ : ϕ(λ) E n E reg n. LL Λ corresponds to the extension C[a 2,..., a n ]/(D LL ) C[f 1,..., f n 1 ]/(D Λ ).

Factorisations of type Λ Theorem (R.) For any stratum Λ in L 2 : LL Λ is a finite morphism of degree (n 2)! hn 1 W deg D Λ ;

Factorisations of type Λ Theorem (R.) For any stratum Λ in L 2 : (n 2)! hn 1 LL Λ is a finite morphism of degree W deg D Λ ; the number of factorisations of c in n 2 reflections + one (length 2) element of conjugacy class corresponding to Λ (in any order) equals : FACT Λ (n 1)! hn 1 n 1 (c) = deg D Λ. W

Submaximal factorisations To obtain FACT n 1 (c), we need to compute Λ L 2 deg(d Λ ).

Submaximal factorisations To obtain FACT n 1 (c), we need to compute Λ L 2 deg(d Λ ). Recall D LL = Λ L 2 D r Λ Λ.

Submaximal factorisations To obtain FACT n 1 (c), we need to compute Λ L 2 deg(d Λ ). Recall D LL = Λ L 2 D r Λ Λ. Proposition Let J LL := Jac(LL) = Jac((a 2,..., a n )/(f 1,..., f n 1 )). Then J LL. = D rλ 1 Λ. Λ L 2

Submaximal factorisations To obtain FACT n 1 (c), we need to compute Λ L 2 deg(d Λ ). Recall D LL = Λ L 2 D r Λ Λ. Proposition Let J LL := Jac(LL) = Jac((a 2,..., a n )/(f 1,..., f n 1 )). Then Corollary J LL. = D rλ 1 Λ. Λ L 2 The number of factorisations of a Coxeter element c in n 1 blocks is :

Conclusion, prospects We travelled from the numerology of Red R (c) (non-ramified part of LL) to that of FACT n 1 (c), without adding any case-by-case analysis.

Conclusion, prospects We travelled from the numerology of Red R (c) (non-ramified part of LL) to that of FACT n 1 (c), without adding any case-by-case analysis. Finer combinatorial formulas, containing new invariant integers for reflection groups (the deg(d Λ )). We recover geometrically some combinatorial results known in the real case [Krattenthaler]. Can we go further (compute the FACT k (c) )? Problem of the complexity of formulas. Should we interpret Chapoton s formula as a ramification formula for LL?

Thank you! (Merci, gracias...)