Sage-Combinat meeting tonight - Bucknell Universitylinux.bucknell.edu/~pm040/Slides/Hivert.pdf ·...

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Sage-Combinat meeting tonight

Sage’s mission:

“To create a viable high-quality and open-source alternativeto 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 startedBring 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. Thiery2,3

1LITIS, Universite Rouen, France

2University of California at Davis, USA

3Laboratoire de Mathematiques d’Orsay, Universite 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 7→ 0): counting tableaux by shape and descents);

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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 MuPADSorry! translation to Sage not yet finished. . .

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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 MuPADSorry! translation to Sage not yet finished. . .

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-Thiery);

• 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-Thiery);

• 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

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Goals of the talk

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A simple example

Definition (Non decreasing parking functions)

f : {1, . . . , n} 7−→ {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 5f (i) 1 1 2 3 5

←→ ???

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A simple example

f : {1, . . . , n} 7−→ {1, . . . , n} is a NDPF if

Catalan objects:

i 1 2 3 4 5f (i) 1 1 2 3 5

←→

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A simple example

f : {1, . . . , n} 7−→ {1, . . . , n} is a NDPF if

Catalan objects:

i 1 2 3 4 5f (i) 1 1 2 3 5

←→

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A simple example

f : {1, . . . , n} 7−→ {1, . . . , n} is a NDPF if

Catalan objects:

i 1 2 3 4 5f (i) 1 1 2 3 5

←→

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A simple example

f : {1, . . . , n} 7−→ {1, . . . , n} is a NDPF if

• f is order preserving i ≤ j =⇒ f (i) ≤ f (j)

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

f : {1, . . . , n} 7−→ {1, . . . , n} is a NDPF if

• f is order preserving i ≤ j =⇒ f (i) ≤ f (j)

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 studyof an algebra / a group / a monoid.

• 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 studyof an algebra / a group / a monoid.

• Gaussian elimination

• endomorphism reduction

• Jordan form . . .

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Crash course intro to representation theory (2)

Definition

A: algebra / group / monoidRepresentation: vector space V with a morphism

ρ : A 7−→ End(V )

(Left) Module: Bilinear operation a.v (for a ∈ A, v ∈ V ) suchthat

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

0 otherwise.

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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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Theorem

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Theorem

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Zoology of monoids

NDPF(P)

biHecke Monoid

0-Hecke Algebra

Regressive Functions

on a Poset

NontrivialGroups

Unitriangular Boolean Matrices

Solomon-Tits Monoid

InverseMonoids

Semilattices

Semigroups

J-Trivial

R-Trivial

L-Trivial

Aperiodic

Ordered

Left Reg. Bands

Trivial Monoid

M1 submonoid of biHecke Monoid

Abelian Groups

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 � onM 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

Proof:

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J -trivial monoid

Proposition

Proof:

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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

Schutzenberger: Aperiodic monoid (xn stabilizes for large n)

Combinatorics !

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Definition

The regular J -classes determine the simple modules.

There can be groups

Combinatorics !

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Definition

The regular J -classes (essentially) determine the simple modules.

There can be groups

Combinatorics !

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Definition

The regular J -classes (essentially) determine the simple modules.

There can be groups

Combinatorics !

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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 indecomposablemodules.

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 iscalled 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

Theorem

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Radical

Definition

Theorem

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Radical

Definition

Theorem

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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

cki ,j ak .

Then writing x =∑

i xiai , one gets for each j ∈ I

xaj =∑i

xiaiaj =∑i ,k

xicki ,jak .

Trace(xaj) =∑u

(xajau|au) =∑i ,k,u

xicki ,jc

uk,u =

∑k,u

cki ,jc

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

cki ,j ak .

Then writing x =∑

i xiai , one gets for each j ∈ I

xaj =∑i

xiaiaj =∑i ,k

xicki ,jak .

Trace(xaj) =∑u

(xajau|au) =∑i ,k,u

xicki ,jc

uk,u =

∑k,u

cki ,jc

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 6= a2}.

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Definition

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

Theorem

Indecomposable projective = decomposition of A itself.

The largest possible modules (every module is the quotient of aprojective).

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Theorem (See e.g. Curtis-Reiner)

Bijection: Simple modules ↔ indecomposable projective modulesDimension 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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Definition

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Definition

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Definition

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Key role of idempotents (converse)

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

∑kj=1 eiej

But eiej ∈ Pj . Direct sum ⇒ eiej =

{ei if i = j

0 else.

Definition

Maximal orthogonal decomposition of 1 into idempotents:

1 =∑

ei eiej = 0 for i 6= 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 forA/rad(A)

• lift the decomposition while keeping orthogonality: Iterate

x := 1− (1− x2)2

until fix point reached (less than dlog2(dim(A))e) 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 7→ xω: the canonical quotient algebra morphism

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

For e ∈ M, inverte =

∑e′≤J e

ge′ .

to get

ge :=∑

e′≤J e

µe′,ee ′ ,

µ : Mobius function of ≤J

Proposition

The family {ge | e ∈ E (M)} is the unique maximal decompositionof 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 := (v0e1−→ v1

e2−→ · · · el−→ vl)

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

• path algebra (category): product = concatenation if last andfirst 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 suchthat 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 arenon-idempotent and

e = lfix(u), rfix(u) = lfix(v), rfix(v) = f .

x ∈ M non idempotent is irreducible if there is no compatiblefactorizations 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 arrowfrom vlfix(x) to vrfix(x).

Sage : generic Algo + examples...

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Combinatorial application

Bijection:

f = 11235 ←→

←→ 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?

Sage-Combinat meeting tonight - Bucknell Universitylinux.bucknell.edu/~pm040/Slides/Hivert.pdf ·...

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