Factorisations of the Garside element in the dual braid monoids

Transcription

1 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

2 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

3 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

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

5 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.

6 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.

7 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;

8 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);

9 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 ),

10 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 ),

11 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 ), where u S v l S (u) + l S (u 1 v) = l S (v);

12 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 ), where u S v l S (u) + l S (u 1 v) = l S (v); A + (W, S) embeds in B(W ) (the braid group of W ).

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

14 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 ).

15 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;

16 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);

17 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 );

18 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 ); M(W, c) embeds in B(W ), but is not isomorphic to the Artin-Tits monoid.

19 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 ); M(W, c) embeds in B(W ), but is not isomorphic to the Artin-Tits monoid. the construction extends to (well-generated) complex reflection groups.

20 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.

21 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. Shephard-Todd s classification (1954): an infinite series with 3 parameters G(de, e, r) ; 34 exceptional groups.

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

23 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 ).

24 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 ). isomorphism : W \V C n v (f 1 (v),..., f n (v)).

25 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

26 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 ).

27 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.)

28 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.) Fundamental example If W = S n (type A), NCP W {noncrossing partitions of an n-gon}.

29 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.) Fundamental example If W = S n (type A), NCP W {noncrossing partitions of an n-gon}. Very rich combinatorial object.

30 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.

31 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 ).

32 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 ).

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

34 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 :

35 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 ;

36 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};

37 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).

38 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).

39 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}

40 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.

41 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. Ex. : FACT n (c) = FACT 1 n(c) = Red R (c). FACT n 1 (c) = FACT n 2.

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

43 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

44 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

45 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.

46 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.

47 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. Basic case in type A : (x i x j ) 2 = Disc(T n σ 1 T n ( 1) n σ n ; T ) 1 i<j n

48 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).

49 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).

50 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.

51 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 }

52 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},

53 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 a n ; f n ). with

54 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 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.

55 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

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

57 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

58 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).

59 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 ).

60 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 ). R-length and conjugacy class of c y,x depend on the position of (y, x) in H...

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

62 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.

63 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.

64 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 ).

65 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.

66 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. Cox-parab(W )/conj. Cox-parab(W ) : parabolic Coxeter elements, i.e. Coxeter elements of a p.s.g. of W.

67 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. Cox-parab(W )/conj. codim(λ) rang(w ) l(c ) Cox-parab(W ) : parabolic Coxeter elements, i.e. Coxeter elements of a p.s.g. of W.

68 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.

69 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)

70 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.)

71 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.

72 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. ω E n, fact induces LL 1 (ω) FACT µ (c), where µ is the composition of ω.

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

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

75 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

76 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 n 2(c)?

77 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

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

79 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}

80 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

81 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.

82 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}.

83 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.

84 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 ).

85 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 ). / deg(ai ) deg(fj ) deg(d LL ) deg(d Λ ) = (n 2)!hn 2 W deg D Λ

86 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 ;

87 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

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

89 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.

90 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

91 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 :

92 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 : ( ) (n 1)! hn 1 (n 1)(n 2) n 1 FACT n 1 (c) = h + d i. W 2 i=1

93 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.

94 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 )).

95 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].

96 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?

97 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? Combinatorics of the block factorisations of the Garside element: is it interesting in other families of Garside structures?

98 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? Combinatorics of the block factorisations of the Garside element: is it interesting in other families of Garside structures?

99 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? Combinatorics of the block factorisations of the Garside element: is it interesting in other families of Garside structures?

100 Thank you! (Merci, gracias...)