Sage-Combinat meeting tonight Sages mission: To create a viable - - PowerPoint PPT Presentation

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Sage-Combinat meeting tonight

Sage’s mission: “To create a viable high-quality and open-source alternative to MapleTM, MathematicaTM, MagmaTM, and MATLABTM” ... “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

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Combinatorial Representation Theory of Algebras: The example of J -trivial monoids

Florent Hivert1 Anne Schilling2 Nicolas M. Thi´ ery2,3

1LITIS, Universit´

e Rouen, France

2University of California at Davis, USA 3Laboratoire 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

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Combinatorial Representation Theory (1)

Representation theory: lots of natural numbers !

• dimension of simple and indecomposable projective modules

(Sn, GLn: Kostka numbers);

• induction and restrictions multiplicities

(Sm × Sn → Sm+n: Littlewood-Richardson rules);

• Cartan invariant matrices and quivers

(Hn(0): counting permutation by descents and recoils);

• decomposition map

(Hn(q → 0): counting tableaux by shape and descents);

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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. . .

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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. . .

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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 . . .

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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 . . .

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Goals of the talk

• show some algorithms in representation theory
• specialization to J -trivial monoids
• get some combinatorics out of it !

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Goals of the talk

• show some algorithms in representation theory
• specialization to J -trivial monoids
• get some combinatorics out of it !

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Goals of the talk

• show some algorithms in representation theory
• specialization to J -trivial monoids
• get some combinatorics out of it !

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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 ← → ???

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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 ← → 1 1 2 2 3 3 4 4 5 5

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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 ← → 1 1 2 2 3 3 4 4 5 5

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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 ← → 1 1 2 2 3 3 4 4 5 5

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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 ∈ NDPFn then so is f ◦ g . NDPFn is a monoid ! Algebra: formal linear combination. This still works if ≤ is replaced by a partial order

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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 ∈ NDPFn then so is f ◦ g . NDPFn is a monoid ! Algebra: formal linear combination. This still works if ≤ is replaced by a partial order

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Crash course intro to representation theory

Basic idea:

assume that we know well enough linear algebra to help the study

• f an algebra / a group / a monoid.

Uses

• Gaussian elimination
• endomorphism reduction
• Jordan form . . .

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Crash course intro to representation theory

Basic idea:

assume that we know well enough linear algebra to help the study

• f an algebra / a group / a monoid.

Uses

• Gaussian elimination
• endomorphism reduction
• Jordan form . . .

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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

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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.

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Example

• Algebra: A = C[NDPFn]
• Space: Vn = Cn

basis: (b1, b2, . . . , bn)

• Action: f .bi := bf (i)

Some submodules : Vk := b1, b2, . . . , bk Some simple modules : Sk = Vk/Vk−1 basis: bk f .bk =

• bk

if f (k) = k

• therwise.

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Pushing the idea further

The regular representation: basis (bm)m∈M Action by multiplication f .bg = bfg. Fact: For NDPFn, the left Cayley graph is acyclic ! Consequence: lots of dimension 1 modules.

Theorem

All irreducible modules up to isomorphism. Warning: there are duplicates. . .

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Pushing the idea further

The regular representation: basis (bm)m∈M Action by multiplication f .bg = bfg. Fact: For NDPFn, the left Cayley graph is acyclic ! Consequence: lots of dimension 1 modules.

Theorem

All irreducible modules up to isomorphism. Warning: there are duplicates. . .

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Pushing the idea further

The regular representation: basis (bm)m∈M Action by multiplication f .bg = bfg. Fact: For NDPFn, the left Cayley graph is acyclic ! Consequence: lots of dimension 1 modules.

Theorem

All irreducible modules up to isomorphism. Warning: there are duplicates. . .

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Zoology of monoids

NDPF(P) biHecke Monoid 0-Hecke Algebra Regressive Functions

• n a Poset

Nontrivial Groups Unitriangular Boolean Matrices Solomon-Tits Monoid

Inverse Monoids

Semilattices

Semigroups

J-Trivial R-Trivial L-Trivial Aperiodic Ordered Basic

Left Reg. Bands Trivial Monoid M1 submonoid of biHecke Monoid Abelian Groups

Bands

Many Rees Semigps

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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).

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The J Green Relation

x ≤J y if and only if x = uyv for some u, v ∈ M.

Definition

Associated equivalence relation xJ 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).

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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.

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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.

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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.

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J -trivial monoid

Proposition

NDPFn is J -trivial. Proof: Define f g iff f (x) ≤ g(x) for all x.

• f (g(x)) ≤ f (x) because g(x) ≤ x and f is order preserving.
• f (g(x)) ≤ g(x)

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Representation theory of monoids

Definition

A J -class is regular iff it contains an idempotent (ie. x2 = x)

Theorem (See e.g. Ganyushkin, Mazorchuk, Steinberg 07)

The regular J -classes determine the simple modules. There can be groups Sch¨ utzenberger: Aperiodic monoid (xn stabilizes for large n)

Combinatorics !

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Representation theory of monoids

Definition

A J -class is regular iff it contains an idempotent (ie. x2 = x)

Theorem (See e.g. Ganyushkin, Mazorchuk, Steinberg 07)

The regular J -classes determine the simple modules. There can be groups Sch¨ utzenberger: Aperiodic monoid (xn stabilizes for large n)

Combinatorics !

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Representation theory of monoids

Definition

A J -class is regular iff it contains an idempotent (ie. x2 = x)

Theorem (See e.g. Ganyushkin, Mazorchuk, Steinberg 07)

The regular J -classes (essentially) determine the simple modules. There can be groups Sch¨ utzenberger: Aperiodic monoid (xn stabilizes for large n)

Combinatorics !

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Representation theory of monoids

Definition

A J -class is regular iff it contains an idempotent (ie. x2 = x)

Theorem (See e.g. Ganyushkin, Mazorchuk, Steinberg 07)

The regular J -classes (essentially) determine the simple modules. There can be groups Sch¨ utzenberger: Aperiodic monoid (xn stabilizes for large n)

Combinatorics !

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Representation theory of algebras (building blocks)

Definition

The direct sum of two modules is itself a module U ⊕ V : a.(u ⊕ v) = a.u ⊕ a.v . Every submodule can be written as direct sum of indecomposable modules.

Definition

Indecomposable module: V cannot be written as V = V1 ⊕ V2

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Semi-simplicity

Clearly: irreducible ⇒ indecomposable

Definition

An algebra such that every indecomposable module is irreducible is called semi-simple. This is measured by the so-called radical

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Radical

Definition

Ideal of A: subspace I ⊂ A such that AIA = I (note:Left/Right). Nilpotent Ideal: I n = {0} for large n. Radical rad(A): The largest nilpotent ideal.

Theorem

rad(A) is the smallest ideal such that A/rad(A) is semi-simple. A/rad(A) has the same simple modules as A.

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Radical

Definition

Ideal of A: subspace I ⊂ A such that AIA = I (note:Left/Right). Nilpotent Ideal: I n = {0} for large n. Radical rad(A): The largest nilpotent ideal.

Theorem

rad(A) is the smallest ideal such that A/rad(A) is semi-simple. A/rad(A) has the same simple modules as A.

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Radical

Definition

Ideal of A: subspace I ⊂ A such that AIA = I (note:Left/Right). Nilpotent Ideal: I n = {0} for large n. Radical rad(A): The largest nilpotent ideal.

Theorem

rad(A) is the smallest ideal such that A/rad(A) is semi-simple. A/rad(A) has the same simple modules as A.

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Radical

Definition

Ideal of A: subspace I ⊂ A such that AIA = I (note:Left/Right). Nilpotent Ideal: I n = {0} for large n. Radical rad(A): The largest nilpotent ideal.

Theorem

rad(A) is the smallest ideal such that A/rad(A) is semi-simple. A/rad(A) has the same simple modules as A.

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Computing the radical

Theorem (Dickson 1923)

Suppose A is of characteristic 0. Then rad(A) = {x | for all y ∈ xA, Trace(y) = 0} Note: On can also use Ax or AxA. Same idea works for non zero characteristic

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Computing the radical

Theorem (Dickson 1923)

Suppose A is of characteristic 0. Then rad(A) = {x | for all y ∈ xA, Trace(y) = 0} Note: On can also use Ax or AxA. Same idea works for non zero characteristic

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Computing the radical (2)

Choose a basis (ai)i∈I of A, and suppose that aiaj =

• k

ck

i,j ak .

Then writing x =

i xiai, one gets for each j ∈ I

xaj =

• i

xiaiaj =

• i,k

xick

i,jak .

Trace(xaj) =

• u

(xajau|au) =

• i,k,u

xick

i,jcu k,u =

• i

 

k,u

ck

i,jcu k,u

  xi This is a linear system of |I| equations in (xi)i∈I !

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Computing the radical (2)

Choose a basis (ai)i∈I of A, and suppose that aiaj =

• k

ck

i,j ak .

Then writing x =

i xiai, one gets for each j ∈ I

xaj =

• i

xiaiaj =

• i,k

xick

i,jak .

Trace(xaj) =

• u

(xajau|au) =

• i,k,u

xick

i,jcu k,u =

• i

 

k,u

ck

i,jcu k,u

  xi This is a linear system of |I| equations in (xi)i∈I !

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The radical of the algebra of a J -trivial monoid

xω := xn for large n

Theorem

If M a J -trivial monoid, then

• rad(C[M]) is spanned by {ab − ba | a, b ∈ M}.
• rad(C[M]) has for basis {a − aω | a = a2}.

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Representation theory of algebras (building blocks)

Definition

Projective module: V ⊕ · · · = A ⊕ · · · ⊕ A

Theorem

Indecomposable projective = decomposition of A itself. The largest possible modules (every module is the quotient of a projective).

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Representation theory of algebras (building blocks)

Theorem (See e.g. Curtis-Reiner)

Bijection: Simple modules ↔ indecomposable projective modules Dimension formula: dim(A) =

• i∈I

dim(Si) dim(Pi).

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Key role of idempotents

Definition

a ∈ A is idempotent if a2 = a Two idempotents a and b are orthogonal if ab = ba = 0

• 1. (1 − e)2 = 1 − 2e + e2 = 1 − 2e + e = 1 − e is an idempotent
• 2. e and (1 − e) are orthogonal
• 3. consequence: A = Aa ⊕ A(1 − e) ,

therefore Ae is a projective module

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Key role of idempotents

Definition

a ∈ A is idempotent if a2 = a Two idempotents a and b are orthogonal if ab = ba = 0

• 1. (1 − e)2 = 1 − 2e + e2 = 1 − 2e + e = 1 − e is an idempotent
• 2. e and (1 − e) are orthogonal
• 3. consequence: A = Aa ⊕ A(1 − e) ,

therefore Ae is a projective module

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Key role of idempotents

Definition

a ∈ A is idempotent if a2 = a Two idempotents a and b are orthogonal if ab = ba = 0

• 1. (1 − e)2 = 1 − 2e + e2 = 1 − 2e + e = 1 − e is an idempotent
• 2. e and (1 − e) are orthogonal
• 3. consequence: A = Aa ⊕ A(1 − e) ,

therefore Ae is a projective module

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Key role of idempotents

Definition

a ∈ A is idempotent if a2 = a Two idempotents a and b are orthogonal if ab = ba = 0

• 1. (1 − e)2 = 1 − 2e + e2 = 1 − 2e + e = 1 − e is an idempotent
• 2. e and (1 − e) are orthogonal
• 3. consequence: A = Aa ⊕ A(1 − e) ,

therefore Ae is a projective module

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Key role of idempotents (converse)

Suppose that A = P1 ⊕ P2 ⊕ · · · ⊕ Pk. Expands 1 = e1 + e2 + · · · + ek. Then ei1 = ei1 = k

j=1 eiej

But eiej ∈ Pj. Direct sum ⇒ eiej =

• ei

if i = j else.

Definition

Maximal orthogonal decomposition of 1 into idempotents: 1 =

• ei

eiej = 0 for i = j No ei can be written as a sum ei = ei′ +ei′′ with ei′, ei′′ orthogonal.

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Computing a max. orthog. dec. of 1 into idempotents

• compute the center of A/rad(A)
• simultaneous diagonalization gives a decomposition for

A/rad(A)

• lift the decomposition while keeping orthogonality: Iterate

x := 1 − (1 − x2)2 until fix point reached (less than ⌈log2(dim(A))⌉) iterations.

• keep orthogonality

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The ⋆ product and the semi-simple quotient

E(M): set of idempotent of M

Theorem

For x, y ∈ E(M), define x ⋆ y := (xy)ω Then ≤J restricted to E(M) is a lower semi-lattice such that x ∧J y = x ⋆ y As a consequence (M, ⋆) is a commutative monoid

Corollary

Then (C[E(M)], ⋆) is isomorphic to C[M]/rad(C[M]) x → xω: the canonical quotient algebra morphism

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• max. orthog. dec. of 1 for J -trivial monoids

For e ∈ M, invert e =

• e′≤J e

ge′ . to get ge :=

• e′≤J e

µe′,ee′ , µ : M¨

• bius function of ≤J

Proposition

The family {ge | e ∈ E(M)} is the unique maximal decomposition

• f the identity into orthogonal idempotents for CE(M).

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The path algebra of a Quiver

Definition

• Quiver: (edge labeled) graph Q = (V , E)
• path of length l (possibly = 0)

p := (v0

e1

− → v1

e2

− → · · ·

el

− → vl) such that ei is an edge from vi−1 to vi.

• path algebra (category): product = concatenation if last and

first vertex matches else 0.

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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.

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Automorphism sub-monoids

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)).

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Factorizations

Definition

Let x ∈ M non idempotent and e := lfix(x) and f := rfix(x). A factorization x = uv is compatible if u and v are non-idempotent and e = lfix(u), rfix(u) = lfix(v), rfix(v) = f . x ∈ M non idempotent is irreducible if there is no compatible factorizations x = uv.

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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 ve for each idempotent e of the monoid;
• For each irreducible element x in the monoid there is an arrow

from vlfix(x) to vrfix(x). Sage : generic Algo + examples...

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Combinatorial application

Bijection: f = 11235 ← → 1 1 2 2 3 3 4 4 ← → lfix(f ) rfix(f ) (1, 2, 4) (2, 3, 4) For 0-Hecke algebra : combinatorial description of the quiver (improve Duchamp-H.-Thibon, Fayers).

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Work in progress

• Finding good idempotents is hard (see Hn(0): Denton)

Do we really need them for Cartan invariants, quiver ?

• R-trivial monoids and DA:

Pure combinatorics (graph theory + counting element)

• Aperiodic monoids:

Small Gaussian elimination over Q (actually Z)

• Is the q-Cartan matrix combinatorial?