The Coxeter complex and the Euler characteristics of a Hecke algebra

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1 The Coxeter complex and the Euler characteristics of a Hecke algebra Tommaso Terragni A joint work with Thomas Weigel Università di Milano Bicocca Ischia Group Theory 2012 T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

2 Coxeter groups A Coxeter system (W, S) consists of a group W generated by a finite set S of involutions subject to relations of the form (st) ms,t, for s t S and suitable m s,t Z 2 { }. For I S the parabolic subsystem is (W I, I ), where W I = I. Let l: W N 0 be the length function w.r.t. the generating system S. Define the Poincaré series of (W, S) as p (W,S) (t) = w W t l(w) Z t. It is well-known that p (W,S) (t) is a rational function and, for W =, 1 p (W,S) (t) = I S ( 1) S\I 1 1 p (WI,I )(t). ( ) T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

3 Euler characteristic of Coxeter groups Theorem (Serre 71) If (W, S) is a Coxeter group then W is of type WFL (hence VFP): thus, it admits an Euler characteristic χ W. Moreover, χ W = 1 p (W,S) (1). T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

4 Euler characteristic of Coxeter groups Theorem (Serre 71) If (W, S) is a Coxeter group then W is of type WFL (hence VFP): thus, it admits an Euler characteristic χ W. Moreover, χ W = 1 p (W,S) (1). A recursive, alternating-sum formula (like ( )) is often associated to the Euler characteristic of a suitably defined algebraic object. Thus, the following question is natural: Main problem Does there exist an object (associated with (W, S)) such that its Euler characteristics coincides with the inverse of the Poincaré series p (W,S) (t)? What is a sensible definition of Euler characteristic for such objects? T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

5 Hecke algebras Datum: a Coxeter system (W, S), a commutative ring R and a parameter q R. R-Hecke algebra H q (W, S) Let H be the free R-module with basis { T w w W } and associative R-linear multiplication defined by { T sw if l(sw) > l(w) T st w = qt sw + (q 1)T w if l(sw) < l(w), for s S and w W. The function l is the length function of (W, S). T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

6 Hecke algebras Datum: a Coxeter system (W, S), a commutative ring R and a parameter q R. R-Hecke algebra H q (W, S) Let H be the free R-module with basis { T w w W } and associative R-linear multiplication defined by { T sw if l(sw) > l(w) T st w = qt sw + (q 1)T w if l(sw) < l(w), for s S and w W. The function l is the length function of (W, S). For q = 1 R, one has H 1(W, S) R[W ]. Thus, a Hecke algebra can be considered as a generic form or a deformation of the group algebra of the Coxeter group. T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

7 Hecke algebras Datum: a Coxeter system (W, S), a commutative ring R and a parameter q R. R-Hecke algebra H q (W, S) Let H be the free R-module with basis { T w w W } and associative R-linear multiplication defined by { T sw if l(sw) > l(w) T st w = qt sw + (q 1)T w if l(sw) < l(w), for s S and w W. The function l is the length function of (W, S). For q = 1 R, one has H 1(W, S) R[W ]. Thus, a Hecke algebra can be considered as a generic form or a deformation of the group algebra of the Coxeter group. Hecke algebras (and their parabolic subalgebras) are canonically equipped with A free R-basis {T w w W }, a linear character ε q : H R, ε q(t w ) = q l(w) (trivial module R q), an antipodal map : H op H, (T w ) = T w 1, a poset of parabolic subalgebras H I. T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

8 Euler algebras, 1 We propose the definition of Euler algebras to single out the sufficient conditions for an (associative) algebra to admit a notion of Euler characteristic. This mimicks the behaviour of group algebras. T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

9 Euler algebras, 1 We propose the definition of Euler algebras to single out the sufficient conditions for an (associative) algebra to admit a notion of Euler characteristic. This mimicks the behaviour of group algebras. Definition Euler R-algebra Let A be an associative R-algebra, together with be a free R-basis B 1, an antipodal map : H op H such that B = B, an augmentation ε: A R onto a one-dimensional left A-module R ε, such that ε = ε, a canonical trace function µ: A/[A, A] R defined by µ(a) = δ 1,aε(a) for a B. Suppose further that R q is a left A-module of type FP, i.e., it admits a finite, projective A-resolution. Then, the 5-tuple A = (A, B,, ε, µ) is an Euler algebra. T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

10 Euler algebras, 2 Let M be a left A-module of type FP (with a finite, projective resolution P ). The Hattori Stallings rank of M is defined as the element r M = k 0( 1) k p k(p k ) + [A, A] A/[A, A], where γ(pk p k ) = id Pk for the natural isomorphism γ : Pk P k End A (P k ). This only depends on the module M, and not on P. If B A is an inclusion of Euler algebras and if A is B-flat, then, for B-module M r ind A B M = r M + [A, A]. T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

11 Euler algebras, 2 Let M be a left A-module of type FP (with a finite, projective resolution P ). The Hattori Stallings rank of M is defined as the element r M = k 0( 1) k p k(p k ) + [A, A] A/[A, A], where γ(pk p k ) = id Pk for the natural isomorphism γ : Pk P k End A (P k ). This only depends on the module M, and not on P. If B A is an inclusion of Euler algebras and if A is B-flat, then, for B-module M r ind A B M = r M + [A, A]. A trace function is a map of R-modules µ: A/[A, A] R. Definition (Euler characteristic) For an R-Euler algebra A = (A, B,, ε, µ), define χ A = µ(r Rε ). T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

12 rstr stsr str srtr rtsr tsr tr rtr tsrt sr r rt srt trsr rsr 1 t rsrt rs s st rst strs trs ts sts trst rtrs rts rsts srts Hecke algebras are Euler, 1 We already know that Hecke algebras admit a free basis {T w w W }, an antipode T w T w 1, an augmentation ε q and a canonical trace function µ defined by µ(t w ) = δ 1,w. Last step: proving that the trivial module R q is of type FP. Coxeter groups have a very rich geometry, and from the Coxeter complex one can produce resolutions of the trivial module. T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

13 Hecke algebras are Euler, 1 We already know that Hecke algebras admit a free basis {T w w W }, an antipode T w T w 1, an augmentation ε q and a canonical trace function µ defined by µ(t w ) = δ 1,w. Last step: proving that the trivial module R q is of type FP. Coxeter groups have a very rich geometry, and from the Coxeter complex one can produce resolutions of the trivial module. Unfortunately, there is no obvious geometry for Hecke algebras, T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

14 Hecke algebras are Euler, 1 We already know that Hecke algebras admit a free basis {T w w W }, an antipode T w T w 1, an augmentation ε q and a canonical trace function µ defined by µ(t w ) = δ 1,w. Last step: proving that the trivial module R q is of type FP. Coxeter groups have a very rich geometry, and from the Coxeter complex one can produce resolutions of the trivial module. Unfortunately, there is no obvious geometry for Hecke algebras, but we constructed a module-theoretic analogue C of the Coxeter complex Σ (canonical up to the choice of a total order < on S). Let I S and denote ind S I the induction from left H I - to left H-modules. If W I is finite and p (W,S) (q) R is invertible, then let e I = w W I T w. Then 1 p (W,S) (q) e 2 I = e I and ind S I R q He I. Moreover the Hattori Stallings rank of ind S I R q is e I + [H I, H I ]. T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

15 Hecke algebras are Euler, 2 For I S and s S, define deg(i ) = S I 1 and the sign map sgn(s, I ) = ( 1) { t S\I t<s }. The Hecke Coxeter complex (C, ). S 1 (C, ) = 0 C S 1 C S C 1 C 0 0, where C k = deg(i )=k inds I R q and k (T w η I ) = sgn(s, I )ε q(t wj )T w J η J. s S\I J=I {s} T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

16 Hecke algebras are Euler, 2 For I S and s S, define deg(i ) = S I 1 and the sign map sgn(s, I ) = ( 1) { t S\I t<s }. The Hecke Coxeter complex (C, ). S 1 (C, ) = 0 C S 1 C S C 1 C 0 0, where C k = deg(i )=k inds I R q and k (T w η I ) = sgn(s, I )ε q(t wj )T w J η J. s S\I J=I {s} Theorem A Let (W, S) be a Coxeter group with 2 S < and let C be the Hecke Coxeter complex of the R-Hecke algebra H = H q(w, S). If (W, S) is spherical, then H k (C) = 0 unless k = 0 or k = S 1. Moreover, H 0(C) R q and H S 1 (C) R 1. If (W, S) is non-spherical then C is acyclic with H 0(C) R q. T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

17 Euler characteristic of Hecke algebras Suppose p (WI,I )(q) is invertible for all I such that W I is finite. Then, by a standard argument, the trivial module R q is of type FP. ( ) T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

18 Euler characteristic of Hecke algebras Suppose p (WI,I )(q) is invertible for all I such that W I is finite. Then, by a standard argument, the trivial module R q is of type FP. ( ) Proposition B Any Hecke algebra H q(w, S) is of type FP if the condition ( ) on q holds true. With the canonical structure H = (H,, ε q, B, µ) described before, any Hecke algebra satisfying ( ) is an Euler algebra. When R = Z q and W = one thus has χ H = I S ( 1) S\I 1 1 p (WI,I )(q) = 1 p (W,S) (q). T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

19 Euler characteristic of Hecke algebras Suppose p (WI,I )(q) is invertible for all I such that W I is finite. Then, by a standard argument, the trivial module R q is of type FP. ( ) Proposition B Any Hecke algebra H q(w, S) is of type FP if the condition ( ) on q holds true. With the canonical structure H = (H,, ε q, B, µ) described before, any Hecke algebra satisfying ( ) is an Euler algebra. When R = Z q and W = one thus has χ H = I S ( 1) S\I 1 1 p (WI,I )(q) = 1 p (W,S) (q). Dealing separately also the (elementary) case W < one has Corollary If R = Z q and H is the R-Hecke algebra with Coxeter system (W, S), then χ H p (W,S) (q) = 1 T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10

20 References H. Bass, Euler characteristics and characters of discrete groups N. Bourbaki, Groupes et algèbres de Lie, Ch. 4 à 6. K.S. Brown, Cohomology of groups. M. Geck & G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras. J.-P. Serre, Cohomologie des groupes discrets. T. Terragni & T.S. Weigel, The Coxeter complex and the Euler characteristic of a Hecke algebra, ArXiv:math/ Picture of (2, 3, 7) hyperbolic tiling by C. Rocchini, under CC-BY 2.5, heptakis heptagonal tiling.png T. Terragni (UniMiB) Euler characteristic of Hecke algebras IGT / 10