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Tutte polynomials Hyperplane arrangements Hyperplane arrangements Computing Tutte polynomials Arithmetic Tutte polynomials Hypertoric arrangements Computing Tutte polynomials Computing Tutte polynomials

Polinomios aritméticos de Tutte

Federico Ardila

San Francisco State UniversityUniversidad de Los Andes, Bogotá, Colombia.

Congreso Colombiano de MatemáticasBarranquilla, 16 de julio de 2013


Outline

1. Tutte polynomials2. Hyperplane arrangements3. Computing Tutte polynomials4. Corte de comerciales.5. Arithmetic Tutte polynomials6. Toric arrangements7. Computing arithmetic Tutte polynomials

Joint work with:Federico Castillo (U. de Los Andes + U. of California at Davis)Mike Henley (San Francisco State University)


1. THE TUTTE POLYNOMIAL.

Let A ⊆ Kn be a collection of vectors.

The Tutte polynomial of A is

TA(x , y) =∑B⊆A

(x − 1)r(A)−r(B)(y − 1)|B|−r(B).

where, for each B ⊆ A, the rank of B is

r(B) = dim span(B)

Example: C2 = {2e1,2e2,e1 + e2,e1 − e2} ⊆ R2


subset rank contribution∅ 0 (x − 1)2(y − 1)0

a,b, c,d 1 (x − 1)1(y − 1)0

ab,ac,ad ,bc,bd , cd 2 (x − 1)0(y − 1)0

abc,abd ,acd ,bcd 2 (x − 1)0(y − 1)1

abcd 2 (x − 1)0(y − 1)2

T (x , y) = (x − 1)2 + 4(x − 1) + 6 + 4(y − 1) + (y − 1)3

= x2 + y2 + 2x + 2y

Ojo. Need Char K 6= 2.


WHY CARE ABOUT THE TUTTE POLYNOMIAL?

Many important invariants of A are evaluations of TA(x , y).

For vector arrangements:• T (1,1) = number of bases.• T (2,1) = number of independent sets.• T (1,2) = number of spanning sets.

These are nice, but maybe not terribly interesting.


Many invariants of A are evaluations of TA(x , y).

For graphs:From a graph G = (V ,E) I get a vector arrangement AG ∈ KV :

AG = {ei − ej : ij is an edge of G}

• T (1,1) = number of spanning trees.• T (2,0) = number of acyclic orientations of edges.• T (0,2) = number of totally cyclic orientations of edges.• (−1)v−c qc T (1− q,0) = chromatic polynomial = number

of proper q-colorings of the vertices.• (−1)e−v+c T (0,1− t) = flow polynomial = number of

nowhere zero t-flows of the edges.

[Stanley, 1973, 1980] [Tutte, 1947] [Crapo, 1969]


2. HYPERPLANE ARRANGEMENTS

For hyperplane arrangements:

Vector a ∈ Kn 7→ Hyperplane Ha = {x ∈ (Kn)∗ : a · x = 0}.

Vector arr. A ⊆ Kn 7→ Hyperplane arr. HA = {Ha : a ∈ A}Complement V (A) = Kn \

⋃H∈AH

Example.C2 : 2x = 0, x + y = 0, 2y = 0, x − y = 0


Example. C3

Root system:• ± ei (1 ≤ i ≤ 3)• ± ei ± ej (1 ≤ i < j ≤ 3)

Hyperplanes:2x = 0,2y = 0,2z = 0

x + y = 0, y + z = 0, z + x = 0x − y = 0, y − z = 0, z − x = 0


Many important invariants of A are evaluations of TA(x , y).

For hyperplane arrangements:

• (K = R)

(−1)nT (2,0) = number of regions of V (A)

[Zaslavsky, 1975]

• (K = C)

T (1− q,0) =∑

i

dim H i(V (A);Z)(−q)i

[Orlik and Solomon, 1980, Goresky-MacPherson, 1988]

• (K = Fq)|T (1− q,0)| = |V (A)|

[Crapo and Rota, 1970]


WHY IS THE TUTTE POLYNOMIAL IN SO MANY PLACES?

Given a hyperplane arrangement A and a hyperplane H:

Deletion: A \ H = A \ HContraction: A/H = {H ′ ∩ H : H ′ ∈ A}

A Tutte-Grothendieck invariant is a function which behaveswell under deletion and contraction:

f (A) = f (A \ H) + f (A/H) (H nontrivial)

2

4

1

3

21

3

2 1

Theorem. (Brylawski, 1972) The Tutte polynomial is the universalT-G invariant. Every other one is an evaluation of TA(x , y).


3. COMPUTING TUTTE POLYNOMIALS

Finite field method.

Let χ(q, t) = (t − 1)r T(

q+t−1t−1 , t

).

Theorem. LetA be a Z-arrangement. Let q be a large enoughprime power, and let Aq be the induced arrangement in Fn

q.Then

qn−rχA(q, t) =∑

p∈Fnq

th(p)

where h(p) = number of hyperplanes of Aq that p lies on.

Computing Tutte polynomials is #P-hard, so we cannot expectmiracles from this method. Still, it is often very useful.


An example.

C2: 2x1 = 0, 2x2 = 0, x1 + x2 = 0, x1 − x2 = 0.

χC2(q, t) =

∑p∈Fn

q

th(p)

= t4 + t1[4(q − 1)] + t0[q2 − 4q + 3]


Corte de comerciales.

San Francisco State University – Colombia Combinatorics Initiative

Para más información sobre

• combinatoria enumerativa (el siguiente semestre),• matroides,• politopos,• grupos de Coxeter,• álgebra conmutativa combinatoria, y• álgebras de Hopf en combinatoria,

pueden ver los (200+) videos y las notas de mis cursos deSan Francisco State University y la Universidad de Los Andes:

http://math.sfsu.edu/federico/http://youtube.com/user/federicoelmatematico


Decía que la aritmética...


4. THE ARITHMETIC TUTTE POLYNOMIAL.

Let A ⊆ Zn be a collection of vectors.

The arithmetic Tutte polynomial of A is

TA(x , y) =∑B⊆A

m(B)(x − 1)r(A)−r(B)(y − 1)|B|−r(B).

where, for each B ⊆ A, the rank of B is

r(B) = dim span(B)

and the multiplicity of B is

m(B) = index of ZB inside Zn ∩ span(B)

Example: C2 = {2e1,2e2,e1 + e2,e1 − e2} ⊆ Z2


subset rank multiplicity contribution∅ 0 1 (x − 1)2(y − 1)0

a, c 1 2 (x − 1)1(y − 1)0

b,d 1 1 (x − 1)1(y − 1)0

ab,ad ,bc,bd , cd 2 2 (x − 1)0(y − 1)0

ac 2 4 (x − 1)0(y − 1)0

abc,abd ,acd ,bcd 2 2 (x − 1)0(y − 1)1

abcd 2 2 (x − 1)0(y − 1)2

M(x , y) = 1(x − 1)2 + [2 + 2 + 1 + 1](x − 1) + [4 + 2 + 2+

+2 + 2 + 2] + [2 + 2 + 2 + 2](y − 1) + 2(y − 1)3

= x2 + 2y2 + 4x + 4y + 3


5. HYPERTORIC ARRANGEMENTS

Let T = (K∗)n = (K \ 0)n be an n- torus.

For hypertoric arrangements:

Vector a ∈ Zn 7→ Hypertorus Ta = {x ∈ (Kn)∗ : xa = 1} ⊂ T .

Vector arr. A ⊆ Kn 7→ Hyperplane arr. TA = {Ha : a ∈ A}Complement R(A) = T \

⋃T∈TA T

Example.C2: x2 = 1, xy = 1, y2 = 1, x/y = 1


Several important invariants of A are evaluations of MA(x , y).

For hypertoric arrangements:• (K = R)

(−1)nM(1,0) = number of regions of R(A) in Sn1

[Ehrenborg–Readdy–Sloane 2009, Moci, 2012]

• (K = C)

qnM(2 + 1q ,0) =

∑i

dim H i(R(A);Z)qi

[De Concini–Procesi 2005, Moci, 2012]

• (K = Fq+1, if q ≡ 1(mod N))

(−1)nM(1− q,0) = |R(A)|

[Bränden–Moci 2013, A.–Castillo–Henley 2013]



Geometry:

1. [Stanley 1991] The zonotope of A is

Z (A) = {∑a∈A

λa · a : 0 ≤ λ ≤ 1}

• volume of Z (A) = M(1,1)

• number of lattice points of Z (A) = M(2,1)

• number of interior lattice points of Z (A) = M(0,1)



Box spline theory:

Numerical analysis⋂

Index theory⋂

Algebraic combinatorics

The Hilbert series of the Dahmen – Micchelli space and theDe Concini – Procesi – Vergne space are:

• Hilb(DM(A); q) = qnM( 1q ,1). [Dahmen–Micchelli 1985]

• Hilb(DPV (A); q) = qnM(1 + 1q ,1) [A. 2012]


6. COMPUTING ARITHMETIC TUTTE POLYNOMIALS

Finite field method.

Let Ψ(q, t) = (t − 1)r M(

q+t−2t−1 , t

).

Theorem. [A. – Castillo – Henley 2012, Bränden – Moci 2012]Let A be a toric arrangement. Let q be a large enough prime withq ≡ 1(mod N), and let Aq be the induced arrangement in (F∗q)n.Then

qn−r ΨA(q, t) =∑

p∈ (F∗q )n

th(p)

where h(p) = number of hypertori of Aq that p lies on.


ROOT SYSTEMS

Root systems are arguably the most important vectorconfigurations in mathematics. They are crucial in theclassification of regular polytopes, simple Lie groups and Liealgebras, cluster algebras, etc.

“Classical root systems":

A+n = {ei − ej : 1 ≤ i ≤ j ≤ n}

B+n = {ei ± ej : 1 ≤ i ≤ j ≤ n} ∪ {ei : 1 ≤ i ≤ n}

C+n = {ei ± ej : 1 ≤ i ≤ j ≤ n} ∪ {2ei : 1 ≤ i ≤ n}

D+n = {ei ± ej : 1 ≤ i ≤ j ≤ n}


TUTTE POLYNOMIALS OF CLASSICAL ROOT SYSTEMS

We compute the (arithmetic) Tutte polynomials of An,Bn,Cn,Dn

Finite field method: Compute a(n) (arithmetic) Tuttepolynomial by solving a counting problem over a finite field.

Example:

Cn : x2i = 1, xixj = 1, xi/xj = 1

Count points (p1, . . . ,pn) ∈ (F∗q)n by the number of equationsthey satisfy. (Basic number theory, quadratic residues,...)



The answers are best expressed in terms of the (arithmetic)Tutte generating functions ΨA,ΨB,ΨC ,ΨD:

ΨA(x , y , z) =∑n≥0

ΨAn (x , y)zn

n!

and the two-variable Rogers-Ramanujan function:

F (α, β) =∑n≥0

αnβ(n2)

n!

Motivating example:

Theorem. [A. 2002] The Tutte generating function for the type Aroot systems is:

ΨA(x , y , z) = F (z, y)x .

Similar (more complicated) formulas hold for types B,C, and D.



Theorem. [A.–Castillo–Henley 2013] The arithmetic Tuttegenerating functions for the classical root systems are:

ΨB = F (2Z ,Y )X4−1F (Z ,Y 2)F (YZ ,Y 2)

[F (2Z ,Y )

X4 + F (−2Z ,Y )

X4

]ΨC = F (2Z ,Y )

X2−1F (YZ ,Y 2)2

ΨD = F (2Z ,Y )X4−1F (Z ,Y 2)2

[F (2Z ,Y )

X4 + F (−2Z ,Y )

X4

]and

ΨA =∑n∈N

ϕ(n)([

F (Z ,Y )F (ωnZ ,Y )F (ω2nZ ,Y ) · · ·F (ωn−1

n Z ,Y )]X/n − 1

)where ϕ(n) = #{m ∈ N : 1 ≤ m ≤ n, (m,n) = 1} is Euler’stotient function and ωn is a primitive nth root of unity for each n.

Corollary. Formulas for zonotopes, DM and DPV-spaces, etc.


muchas gracias

El artículo está en:

http://arxiv.org/abs/1305.6621http://math.sfsu.edu/federico

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