1 / 39
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
2 / 39
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
3 / 39
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);
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 MuPADSorry! 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 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 . . .
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 . . .
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
Goals of the talk
• show some algorithms in representation theory
• specialization to J -trivial monoids
• get some combinatorics out of it !
7 / 39
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
←→ ???
7 / 39
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
←→
11
2
2
3
3
4
4
5
5
7 / 39
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
←→
11
2
2
3
3
4
4
5
5
7 / 39
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
←→
11
2
2
3
3
4
4
5
5
8 / 39
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
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
8 / 39
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
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
9 / 39
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.
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 studyof an algebra / a group / a monoid.
Uses
• Gaussian elimination
• endomorphism reduction
• Jordan form . . .
10 / 39
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
11 / 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.
12 / 39
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.
13 / 39
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. . .
13 / 39
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. . .
13 / 39
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. . .
14 / 39
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
Basic
Left Reg. Bands
Trivial Monoid
M1 submonoid of biHecke Monoid
Abelian Groups
Bands
Many Rees Semigps
15 / 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).
16 / 39
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).
17 / 39
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.
17 / 39
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.
17 / 39
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.
18 / 39
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)
19 / 39
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 !
19 / 39
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 !
19 / 39
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
Schutzenberger: Aperiodic monoid (xn stabilizes for large n)
Combinatorics !
19 / 39
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
Schutzenberger: Aperiodic monoid (xn stabilizes for large n)
Combinatorics !
20 / 39
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 indecomposablemodules.
Definition
Indecomposable module: V cannot be written as V = V1 ⊕ V2
21 / 39
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
22 / 39
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.
22 / 39
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.
22 / 39
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.
22 / 39
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.
23 / 39
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
23 / 39
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
24 / 39
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 =
∑i
∑k,u
cki ,jc
uk,u
xi
This is a linear system of |I | equations in (xi )i∈I !
24 / 39
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 =
∑i
∑k,u
cki ,jc
uk,u
xi
This is a linear system of |I | equations in (xi )i∈I !
25 / 39
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}.
26 / 39
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 aprojective).
27 / 39
Representation theory of algebras (building blocks)
Theorem (See e.g. Curtis-Reiner)
Bijection: Simple modules ↔ indecomposable projective modulesDimension formula:
dim(A) =∑i∈I
dim(Si ) dim(Pi ).
28 / 39
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
28 / 39
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
28 / 39
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
28 / 39
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
29 / 39
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.
30 / 39
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
31 / 39
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
32 / 39
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).
33 / 39
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.
34 / 39
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.
35 / 39
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)).
36 / 39
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 .
37 / 39
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...
38 / 39
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).
39 / 39
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?
Top Related