Sorting monoids on Coxeter groups

Transcription

1 Sorting monoids on Coxeter groups A computer exploration with Sage-Combinat Florent Hivert 1 Anne Schilling 2 Nicolas M Thiéry 2,3 1 LITIS/LIFAR, Université Rouen, France 2 University of California at Davis, USA 3 Laboratoire de Mathématiques d Orsay, Université Paris Sud, France LIPN, Monday 23rd of 2009 arxiv: v1 [mathrt] arxiv: v1 [mathrt] arxiv: v1 [mathco] + research in progress 1 / 28

2 Sage-Combinat (combinatsagemathorg) 50+ research articles NSF Sponsored Sage: 300 tickets / 100k lines integrated in Sage MuPAD: 115k lines of MuPAD, 15k lines of C++, 32k lines of tests, 600 pages of doc Nicolas Borie, Daniel Bump, Jason Bandlow, Adrien Boussicault, Vincent Delecroix, Paul-Olivier Dehaye, Tom Denton, Dan Drake, Teresa Gomez Diaz, Mike Hansen, Ralf Hemmecke, Florent Hivert, Brant Jones, Sébastien Labbé, Yann Laigle-Chapuy, Andrew Mathas, Gregg Musiker, Steven Pon, Franco Saliola, Anne Schilling, Mark Shimozono, Nicolas M Thiéry, Justin Walker, Qiang Wang, Mike Zabrocki, 2 / 28

3 Bubble (anti) sort algorithm Underlying combinatorics: right permutohedron / 28

4 Coxeter groups Definition (Coxeter group W ) Generators : s 1, s 2, (simple reflections) Relations: si 2 = 1 and s i s j }{{} = s j s i, for i j }{{} m i,j m i,j Reduced word Length 4 / 28

5 Orders on words and on Coxeter group elements Definition (Orders on words) Let u = u 1 u k and v = v 1 v l : u left factor of v if v = u 1 u k u right factor of v if v = u 1 u k u factor of v if v = u 1 u k u subword of v if v = u 1 u 2 u k Definition (Orders on Coxeter group elements) Right weak order Left weak order Left-right weak order Bruhat order 5 / 28

6 Blocks of permutations Definition (Block of a permutation w) Type A: sub-permutation matrix Type free: J, K such that W J w = ww K Example: w := Simple permutation: cf [Albert, Atkinson 05] {blocks of w}: sub-lattice of the Boolean lattice Definition (HST09: Cutting poset (W, )) u w if u = w J with J block (almost) lattice Möbius function: inclusion-exclusion along minimal blocks 6 / 28

7 Hecke monoid Definition (0-Hecke monoid H 0 (W ) of a Coxeter group W ) Generators : π 1, π 2, (simple reflections) Relations: π 2 i = π i and braid relations Theorem H 0 (W ) = W + lots of nice properties Motivation: simple combinatorial model (bubble sort) appears in Iwahori-Hecke algebras, Schur symmetric functions, Schubert, Kazhdan-Lusztig polynomials, and Macdonald, (affine) Stanley symmetric functions, mathematical physics, Schur-Weyl duality for quantum groups, representations of GL(F q ), 7 / 28

8 The Big Picture NDPF (Bruhat(W )) End(< L (W )) End(BooleanLattice) M 1 π 1, π 2,, π 1, π 2, π 1, π 2,, s 1, s 2, π 0π 1, π 2, H ζ( W ) H 1( W ) H 0( W ) π 0π 1, π 2, H 0( W ) π 0π 1, π 2, W s 0s 1, s 2, H q( W ) H ζ(w ) H 1(W ) H 0(W ) π 1, π 2, H 0(W ) π 1, π 2, HW Q[π 1, π 2,, s 1, s 2, ] Q[π 1, π 2,, π 1, π 2, ] Q[π 0π 1, π 2, ] W s 1, s 2, H q(w ) H ζ(s n) TL n NDPF n NDPF n NDF n S n H q(s n) NDPF B 8 / 28

9 Question The bi-hecke monoid Size of M(W ) = π 1, π 2,, π 1, π 2, M(S n ) = 1, 3, 23, 477, 31103,? How to attack such a problem? Generators and relations? Representation theory? Theorem (HST08) M(W ) admits W simple / indecomposable projective modules Why do we care? M(W ) = dim S w dim P w w W 9 / 28

10 Representation theory of algebras Module: vector space V with a morphism M End(V ) Simple module: V contains no nontrivial submodule Indecomposable module: V cannot be written as V = V 1 V 2 Projective module: V = C[M] C[M] Theorem (See eg Curtis-Reiner) Simple modules indecomposable projective modules Dimension formula, Key role of idempotents: ev projective module If f = uev then fm is isomorphic to a submodule of em 10 / 28

11 Representation theory of monoids Definition (J-(pre)order) x J y iff x = uyv, for some u, v M x, y M are in the same J-class if x J y and y J x A J-class is regular iff it contains an idempotent Theorem (See eg Ganyushkin, Mazorchuk, Steinberg 07) The regular J-classes (essentially) determine the simple modules = Combinatorial Representation Theory 11 / 28

12 Key combinatorial lemma 12 / 28

13 Lemma Key combinatorial lemma 132 For f M(W ) and w W : (s i w)f = wf or s i (wf ) Proof Exchange property / associativity Corollary If w = uv, then (uv)f = u (vf ), where u < B u 13 / 28

14 14 / 28 Bubble sort operators and Coxeter groups Sorting monoids and algebras Combinatorics of the bi-hecke monoid Some elements of the monoid

15 Representation theory of M(W ) Theorem (HST 08) M(W ) admits W simple modules Sketch of proof M acts transitively on intervals [u, v] L The image set of an idempotent is an interval [u, v] L! e w idempotent with image set [1, w] L, for any w W (e w ) w W : transversal of the regular J-classes f = uev if and only if im(f ) is a subinterval of im(e) The groups e w Me w are trivial Problem Dimension of simple and projective modules? 15 / 28

16 4312 Translation algebras Definition (Translation algebra) 1234 T w := Q[π 1, π 2,, π 1, π 2, ] acting on Q[1, w] R Blocks: J = {}, {1, 2}, {3}, {1, 2, 3} = Submodules P J T w : max algebra stabilizing all P J = Repr theory T w quotient of Q[M(W )]; top: simple module S w of M Dimension: inclusion-exclusion along the cutting poset Generating series calculation? 16 / 28

17 Definition The Borel submonoid M 1 Submonoid M 1 := {f M, f (1) = 1} Properties (HST 09) Generated by e w for w grassmanian Weakly increasing and contracting on Bruhat = J-trivial Idempotents: (e w ) w W W simple modules of dimension 1 Semi simple quotient: monoid algebra of (W, L ) Conjugacy order among idempotents: < L dim P w = {f M 1, f (w) = w}? Problem Inducing these results to M? 17 / 28

18 Representation theory of J-trivial monoids Theorem (HST 09) Combinatorial description of: Simple modules Projective modules Cartan matrix Quiver q-cartan matrix (in progress) in term of some statistic on M Question Induction from Borel submonoids? 18 / 28

19 The path algebra of a Quiver Definition Quiver: (edge labeled) graph Q = (V, E) path of length l (possibly = 0) e p := (v 1 e 0 2 e v1 l vl ) such that e i is an edge from v i 1 to v i path algebra (category): product = concatenation if last and first vertex matches else 0 TODO : Add an example 19 / 28

20 Structure theorem for finite dimensional algebras Definition admissible ideal: included in the ideal of path of length 2 Theorem For any (elementary) algebra A, there is a unique quiver Q such that A is the quotient of CQ by an admissible ideal I Elementary algebras: simple module are all 1-dimensional Note: first order approximation of the algebra Note: the ideal I is far from being unique 20 / 28

21 Vertices of the Quiver? Decomposition of the identity: 1 = e f e and f e f e = δ ee f e ; Theorem (HST 09) Association: e M f e C[M], such that Moreover f e = e + smaller terms Theorem The vertices of the Quiver are naturally indexed by the idempotents of the monoid 21 / 28

22 Cartan s invariants Matrix decomposition of the algebra x C[M]: x = e 1,e 2 x e1,e 2 where x e1,e 2 = f e1 xf e2 (xy) e1,e 2 = e x e1,ey e,e2 22 / 28

23 Automorphism sub-monoids and factorizations Definition (Automorphism sub-monoids) raut(x) := {u M xu = x} Proposition There exists a unique idempotent rfix(x) such that raut(x) = {u M rfix(x) J u} Same one the left (laut(x), lfix(x)) 23 / 28

24 Corollary: Cartan invariants + vertex of the quiver Theorem Cartan s invariants: dim(f e1 C[M]f e2 ) = #{x M lfix(x) = e 1 and rfix(x) = e 2 } 24 / 28

25 Factorizations Definition Let x M non idempotent and e := and f := rfix(x) A factorization x = uv is non-trivial if eu e and vf f equivalently if u laut(x) and v raut(x); compatible if u and v are non-idempotent and lfix(u) = e, rfix(v) = f and rfix(u) = lfix(v) ; 25 / 28

26 Factorizations and irreducible Proposition compatible non-trivial; non-trivial and j-minimal compatible Definition x M non idempotent is irreducible if there is no non-trivial factorizations x = uv 26 / 28

27 The Quiver of (the algebra of) a j-trivial monoid Theorem The quiver of the algebra of M is the following: There is one vertex v e for each idempotent e of the monoid; For each irreducible element x in the monoid there is an arrow from v lfix(x) to v rfix(x) Sage : generic Algo + examples 27 / 28

28 Conclusion General strategy: Find combinatorial models for algebras and representations As simple as possible, but no simpler Concrete and effective Use representation theory and computer exploration as a guide Find the right point of view where proofs become trivial Discussions? Integration of Jean-Éric s Semigroupe package into Sage Simple permutations and cutting poset Endomorphisms of the Boolean lattice 28 / 28