Sage-Combinat meeting tonight Sage’s mission: “To create a viable high-quality and open-source alternative to Maple TM , Mathematica TM , Magma TM , and MATLAB TM ” ... “and to foster a friendly community of users and developers” Tonight, Thornton Hall, Room 236 • 7pm-8pm: Introduction to Sage and Sage-Combinat • 8pm-10pm: Help on installation & getting started Bring your laptop! • Design discussions 1 / 39
Combinatorial Representation Theory of Algebras: The example of J -trivial monoids Florent Hivert 1 Anne Schilling 2 ery 2 , 3 Nicolas M. Thi´ 1 LITIS, Universit´ e Rouen, France 2 University of California at Davis, USA 3 Laboratoire de Math´ ematiques d’Orsay, Universit´ e Paris Sud, France San Francisco, August 2010 arXiv:0711.1561v1 [math.RT] arXiv:0912.2212v1 [math.CO] + research in progress 2 / 39
Combinatorial Representation Theory (1) Representation theory: lots of natural numbers ! • dimension of simple and indecomposable projective modules ( S n , GL n : Kostka numbers); • induction and restrictions multiplicities ( S m × S n → S m + n : Littlewood-Richardson rules); • Cartan invariant matrices and quivers ( H n (0): counting permutation by descents and recoils); • decomposition map ( H n ( q �→ 0): counting tableaux by shape and descents); 3 / 39
Combinatorial Representation Theory (2) Mostly effective: computer exploration ! Depending on • the base field ( Q or some extension) • the sparsity of the multiplication table • . . . Dimension up to 50 to 2000. Short demo in MuPAD Sorry! translation to Sage not yet finished. . . 4 / 39
Combinatorial Representation Theory (2) Mostly effective: computer exploration ! Depending on • the base field ( Q or some extension) • the sparsity of the multiplication table • . . . Dimension up to 50 to 2000. Short demo in MuPAD Sorry! translation to Sage not yet finished. . . 4 / 39
Several recent examples are monoid algebras • 0-Hecke algebras (Norton, Carter, Krob-Thibon, Duchamp-H.-Thibon, Fayers, Denton); • Non-decreasing parking function (Denton-H.-Schilling-Thi´ ery); • Solomon-Tits algebras (Schocker, Saliola); • Left Regular Bands (Brown). . . . . . but this fact is seldom used . . . 5 / 39
Several recent examples are monoid algebras • 0-Hecke algebras (Norton, Carter, Krob-Thibon, Duchamp-H.-Thibon, Fayers, Denton); • Non-decreasing parking function (Denton-H.-Schilling-Thi´ ery); • Solomon-Tits algebras (Schocker, Saliola); • Left Regular Bands (Brown). . . . . . but this fact is seldom used . . . 5 / 39
Goals of the talk • show some algorithms in representation theory • specialization to J -trivial monoids • get some combinatorics out of it ! 6 / 39
Goals of the talk • show some algorithms in representation theory • specialization to J -trivial monoids • get some combinatorics out of it ! 6 / 39
Goals of the talk • show some algorithms in representation theory • specialization to J -trivial monoids • get some combinatorics out of it ! 6 / 39
A simple example Definition (Non decreasing parking functions) f : { 1 , . . . , n } �− → { 1 , . . . , n } is a NDPF if • f is order-preserving i ≤ j = ⇒ f ( i ) ≤ f ( j ) • f is regressive: f ( i ) ≤ i Catalan objects: i 1 2 3 4 5 ← → ??? f ( i ) 1 1 2 3 5 7 / 39
A simple example Definition (Non decreasing parking functions) f : { 1 , . . . , n } �− → { 1 , . . . , n } is a NDPF if • f is order-preserving i ≤ j = ⇒ f ( i ) ≤ f ( j ) • f is regressive: f ( i ) ≤ i Catalan objects: 5 4 i 1 2 3 4 5 ← → 3 f ( i ) 1 1 2 3 5 2 1 1 2 3 4 5 7 / 39
A simple example Definition (Non decreasing parking functions) f : { 1 , . . . , n } �− → { 1 , . . . , n } is a NDPF if • f is order-preserving i ≤ j = ⇒ f ( i ) ≤ f ( j ) • f is regressive: f ( i ) ≤ i Catalan objects: 5 4 i 1 2 3 4 5 ← → 3 f ( i ) 1 1 2 3 5 2 1 1 2 3 4 5 7 / 39
A simple example Definition (Non decreasing parking functions) f : { 1 , . . . , n } �− → { 1 , . . . , n } is a NDPF if • f is order-preserving i ≤ j = ⇒ f ( i ) ≤ f ( j ) • f is regressive: f ( i ) ≤ i Catalan objects: 5 4 i 1 2 3 4 5 ← → 3 f ( i ) 1 1 2 3 5 2 1 1 2 3 4 5 7 / 39
A simple example Definition (Non decreasing parking functions) f : { 1 , . . . , n } �− → { 1 , . . . , n } is a NDPF if • f is order preserving i ≤ j = ⇒ f ( i ) ≤ f ( j ) • f is regressive: f ( i ) ≤ i Remark If f , g ∈ NDPF n then so is f ◦ g . NDPF n is a monoid ! Algebra: formal linear combination. This still works if ≤ is replaced by a partial order 8 / 39
A simple example Definition (Non decreasing parking functions) f : { 1 , . . . , n } �− → { 1 , . . . , n } is a NDPF if • f is order preserving i ≤ j = ⇒ f ( i ) ≤ f ( j ) • f is regressive: f ( i ) ≤ i Remark If f , g ∈ NDPF n then so is f ◦ g . NDPF n is a monoid ! Algebra: formal linear combination. This still works if ≤ is replaced by a partial order 8 / 39
Crash course intro to representation theory Basic idea: assume that we know well enough linear algebra to help the study of an algebra / a group / a monoid. Uses • Gaussian elimination • endomorphism reduction • Jordan form . . . 9 / 39
Crash course intro to representation theory Basic idea: assume that we know well enough linear algebra to help the study of an algebra / a group / a monoid. Uses • Gaussian elimination • endomorphism reduction • Jordan form . . . 9 / 39
Crash course intro to representation theory (2) Definition A : algebra / group / monoid Representation : vector space V with a morphism ρ : A �− → End( V ) (Left) Module : Bilinear operation a . v (for a ∈ A , v ∈ V ) such that a . ( b . v ) = ( ab ) . v Define a . v := ρ ( a )( v ), then a . ( b . v ) := ρ ( a )( ρ ( b )( v )) = ( ρ ( a ) ◦ ρ ( b ))( v ) = ρ ( ab )( v ) = ( ab ) . v 10 / 39
Representation theory of algebras (building blocks) Definition Submodule W ⊂ V is a stable subspace (if x ∈ w then a . x ∈ W ). Simple (irreducible) module : no nontrivial submodule. The smallest possible modules. 11 / 39
Example • Algebra: A = C [NDPF n ] • Space: V n = C n basis: ( b 1 , b 2 , . . . , b n ) • Action: f . b i := b f ( i ) Some submodules : V k := � b 1 , b 2 , . . . , b k � Some simple modules : S k = V k / V k − 1 basis: b k � b k if f ( k ) = k f . b k = 0 otherwise. 12 / 39
Pushing the idea further The regular representation: basis ( b m ) m ∈ M Action by multiplication f . b g = b fg . Fact: For NDPF n , the left Cayley graph is acyclic ! Consequence: lots of dimension 1 modules. Theorem All irreducible modules up to isomorphism. Warning: there are duplicates. . . 13 / 39
Pushing the idea further The regular representation: basis ( b m ) m ∈ M Action by multiplication f . b g = b fg . Fact: For NDPF n , the left Cayley graph is acyclic ! Consequence: lots of dimension 1 modules. Theorem All irreducible modules up to isomorphism. Warning: there are duplicates. . . 13 / 39
Pushing the idea further The regular representation: basis ( b m ) m ∈ M Action by multiplication f . b g = b fg . Fact: For NDPF n , the left Cayley graph is acyclic ! Consequence: lots of dimension 1 modules. Theorem All irreducible modules up to isomorphism. Warning: there are duplicates. . . 13 / 39
Zoology of monoids Many Rees Semigps Aperiodic Semigroups biHecke Monoid Basic Left Reg. Bands Regressive R -Trivial Solomon-Tits Functions Monoid on a Poset Bands M1 submonoid of J -Trivial biHecke Monoid 0-Hecke Algebra NDPF(P) Ordered Trivial Monoid Semilattices Unitriangular Inverse Boolean Matrices Monoids Nontrivial Groups L -Trivial Abelian Groups 14 / 39
Green Relations (1951) Definition • x ≤ L y if and only if x = uy for some u ∈ M • x ≤ R y if and only if x = yv for some v ∈ M • x ≤ J y if and only if x = uyv for some u , v ∈ M • x ≤ H y if and only if x ≤ L y and x ≤ R y Reflexive and Transitive but not always antisymmetric (preorder). 15 / 39
The J Green Relation x ≤ J y if and only if x = uyv for some u , v ∈ M . Definition Associated equivalence relation x J y ⇐ ⇒ x ≤ J y and y ≤ J x . J -classes : equivalence classes. A monoid is J -trivial if the associated equivalence relation is trivial (i.e. ≤ J is an order). 16 / 39
J -trivial monoid Proposition A monoid M is J -trivial if and only if there exists an order � on M such that for all x , y ∈ M xy � x and xy � y Proof: ⇒ trivial: take � := ≤ J ⇐ if x ≤ J y then x � y , therefore ≤ J is anti-symmetric. 17 / 39
J -trivial monoid Proposition A monoid M is J -trivial if and only if there exists an order � on M such that for all x , y ∈ M xy � x and xy � y Proof: ⇒ trivial: take � := ≤ J ⇐ if x ≤ J y then x � y , therefore ≤ J is anti-symmetric. 17 / 39
J -trivial monoid Proposition A monoid M is J -trivial if and only if there exists an order � on M such that for all x , y ∈ M xy � x and xy � y Proof: ⇒ trivial: take � := ≤ J ⇐ if x ≤ J y then x � y , therefore ≤ J is anti-symmetric. 17 / 39
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