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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others Accumulation points of real Schur roots Charles Paquette November 22 nd , 2014 CGMRT

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Accumulation points of real Schur roots

Charles Paquette November 22nd, 2014

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Settings

k = ¯ k is an algebraically closed field.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Settings

k = ¯ k is an algebraically closed field. Q = (Q0, Q1) is a connected acyclic quiver with Q0 = {1, 2, . . . , n}.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Settings

k = ¯ k is an algebraically closed field. Q = (Q0, Q1) is a connected acyclic quiver with Q0 = {1, 2, . . . , n}. We may choose an admissible ordering, that is, j → i ∈ Q1 implies i < j.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Settings

k = ¯ k is an algebraically closed field. Q = (Q0, Q1) is a connected acyclic quiver with Q0 = {1, 2, . . . , n}. We may choose an admissible ordering, that is, j → i ∈ Q1 implies i < j. rep(Q) denotes the category of finite dimensional representations of Q over k.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Settings

k = ¯ k is an algebraically closed field. Q = (Q0, Q1) is a connected acyclic quiver with Q0 = {1, 2, . . . , n}. We may choose an admissible ordering, that is, j → i ∈ Q1 implies i < j. rep(Q) denotes the category of finite dimensional representations of Q over k. Given M ∈ rep(Q), we denote by dM ∈ Zn

≥0 its dimension

vector.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Settings

k = ¯ k is an algebraically closed field. Q = (Q0, Q1) is a connected acyclic quiver with Q0 = {1, 2, . . . , n}. We may choose an admissible ordering, that is, j → i ∈ Q1 implies i < j. rep(Q) denotes the category of finite dimensional representations of Q over k. Given M ∈ rep(Q), we denote by dM ∈ Zn

≥0 its dimension

vector. We denote by −, − the Euler-Ringel form of Q, that is, dM, dN = dimkHom(M, N) − dimkExt1(M, N).

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Schur roots

A representation M is Schur if End(M) = k.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Schur roots

A representation M is Schur if End(M) = k. If M is a Schur representation, then dM is a Schur root.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Schur roots

A representation M is Schur if End(M) = k. If M is a Schur representation, then dM is a Schur root. We then call dM        real, if dM, dM = 1; imaginary, if dM, dM ≤ 0; isotropic, if dM, dM = 0; strictly imaginary, if dM, dM < 0;

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Schur roots

∆Q denotes the set of all rays in Rn in the positive orthant.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Schur roots

∆Q denotes the set of all rays in Rn in the positive orthant. We denote by [d] the ray of d ∈ Zn

≥0.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Schur roots

∆Q denotes the set of all rays in Rn in the positive orthant. We denote by [d] the ray of d ∈ Zn

≥0.

A Schur root that is real or isotropic is uniquely determined by its ray.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Schur roots

∆Q denotes the set of all rays in Rn in the positive orthant. We denote by [d] the ray of d ∈ Zn

≥0.

A Schur root that is real or isotropic is uniquely determined by its ray. If d is strictly imaginary, then all integral vectors in [d] are strictly imaginary.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Accumulation points of real roots

This has been studied by C. Hohlweg, J. Labb´ e, V. Ripoll in arXiv:1112.5415.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Accumulation points of real roots

This has been studied by C. Hohlweg, J. Labb´ e, V. Ripoll in arXiv:1112.5415. Another paper by M. Dyer, C. Hohlweg, V. Ripoll in arXiv:1303.6710.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Accumulation points of real roots

This has been studied by C. Hohlweg, J. Labb´ e, V. Ripoll in arXiv:1112.5415. Another paper by M. Dyer, C. Hohlweg, V. Ripoll in arXiv:1303.6710. A third one by C. Hohlweg, J. Pr´ eaux, V. Ripoll in arXiv:1305.0052.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Example 1

Here is an example for ∆Q for Q of type ˜ A2,1

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Example 1

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Example 2

Here is an example for ∆Q for Q : 1 2

3
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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Example 2

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

The canonical decomposition

Theorem (Kac) Every dimension vector can be written as a positive linear combination of Schur roots d1, . . . , dr such that

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

The canonical decomposition

Theorem (Kac) Every dimension vector can be written as a positive linear combination of Schur roots d1, . . . , dr such that ext1(di, dj) = 0 whenever i = j.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

The canonical decomposition

Theorem (Kac) Every dimension vector can be written as a positive linear combination of Schur roots d1, . . . , dr such that ext1(di, dj) = 0 whenever i = j. the coefficient of a strictly imaginary Schur root is one.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

The canonical decomposition

Theorem (Kac) Every dimension vector can be written as a positive linear combination of Schur roots d1, . . . , dr such that ext1(di, dj) = 0 whenever i = j. the coefficient of a strictly imaginary Schur root is one. Derksen and Weyman’s algorithm can be used to find the canonical decomposition of any dimension vector. All is needed is:

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

The canonical decomposition

Theorem (Kac) Every dimension vector can be written as a positive linear combination of Schur roots d1, . . . , dr such that ext1(di, dj) = 0 whenever i = j. the coefficient of a strictly imaginary Schur root is one. Derksen and Weyman’s algorithm can be used to find the canonical decomposition of any dimension vector. All is needed is:

The Euler form −, − of Q.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

The canonical decomposition

Theorem (Kac) Every dimension vector can be written as a positive linear combination of Schur roots d1, . . . , dr such that ext1(di, dj) = 0 whenever i = j. the coefficient of a strictly imaginary Schur root is one. Derksen and Weyman’s algorithm can be used to find the canonical decomposition of any dimension vector. All is needed is:

The Euler form −, − of Q. Know the canonical decomposition of quivers with two vertices.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

The canonical decomposition

Theorem If the canonical decomposition of d ∈ Zn

≥0 involves a strictly

imaginary Schur root, then there exists a small neighborhood of d with the same property.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Rational accumulation points

Corollary If d is a rational accumulation point of real Schur roots, then the canonical decomposition of d involves pairwise orthogonal isotropic Schur roots.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Rational accumulation points

Corollary If d is a rational accumulation point of real Schur roots, then the canonical decomposition of d involves pairwise orthogonal isotropic Schur roots. Theorem If d is an isotropic Schur root, then d is an accumulation point of real Schur roots.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Rational accumulation points

The quiver Q is of weakly hyperbolic type if the symmetrized Euler form has exactly one negative eigenvalue and the others are positive.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Rational accumulation points

The quiver Q is of weakly hyperbolic type if the symmetrized Euler form has exactly one negative eigenvalue and the others are positive. The quiver Q is weakly hyperbolic (or Dynkin or Euclidean) when, for instance:

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Rational accumulation points

The quiver Q is of weakly hyperbolic type if the symmetrized Euler form has exactly one negative eigenvalue and the others are positive. The quiver Q is weakly hyperbolic (or Dynkin or Euclidean) when, for instance:

|Q0| ≤ 3.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Rational accumulation points

The quiver Q is of weakly hyperbolic type if the symmetrized Euler form has exactly one negative eigenvalue and the others are positive. The quiver Q is weakly hyperbolic (or Dynkin or Euclidean) when, for instance:

|Q0| ≤ 3. Q has a full subquiver with n − 1 vertices which is a union of Dynkin quivers.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Rational accumulation points

The quiver Q is of weakly hyperbolic type if the symmetrized Euler form has exactly one negative eigenvalue and the others are positive. The quiver Q is weakly hyperbolic (or Dynkin or Euclidean) when, for instance:

|Q0| ≤ 3. Q has a full subquiver with n − 1 vertices which is a union of Dynkin quivers.

Proposition If Q is weakly hyperbolic, then the rational accumulation points are precisely the isotropic Schur roots of Q.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

Assume that Q is weakly hyperbolic.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

Assume that Q is weakly hyperbolic. The (dimension vectors of the) preprojective representations accumulates to y−.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

Assume that Q is weakly hyperbolic. The (dimension vectors of the) preprojective representations accumulates to y−. The (dimension vectors of the) preinjective representations accumulates to y+.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

Assume that Q is weakly hyperbolic. The (dimension vectors of the) preprojective representations accumulates to y−. The (dimension vectors of the) preinjective representations accumulates to y+. (Ringel) y+, y− are (irrational) eigenvectors of the Coxeter transformation.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

Assume that Q is weakly hyperbolic. The (dimension vectors of the) preprojective representations accumulates to y−. The (dimension vectors of the) preinjective representations accumulates to y+. (Ringel) y+, y− are (irrational) eigenvectors of the Coxeter transformation. Are there other accumulation points?

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

For any (Generalized) Kronecker subcategory C = rep(Q′) of rep(Q), denote by y−

C , y+ C the associated eigenvalues of the

Coxeter transformation of Q′, where y−

C = y+ C when Q′ is

tame.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

For any (Generalized) Kronecker subcategory C = rep(Q′) of rep(Q), denote by y−

C , y+ C the associated eigenvalues of the

Coxeter transformation of Q′, where y−

C = y+ C when Q′ is

tame. Then y−

C , y+ C are accumulation points in ∆Q, irrational if C is

wild.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

For any (Generalized) Kronecker subcategory C = rep(Q′) of rep(Q), denote by y−

C , y+ C the associated eigenvalues of the

Coxeter transformation of Q′, where y−

C = y+ C when Q′ is

tame. Then y−

C , y+ C are accumulation points in ∆Q, irrational if C is

wild. Are there any other?

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

For any (Generalized) Kronecker subcategory C = rep(Q′) of rep(Q), denote by y−

C , y+ C the associated eigenvalues of the

Coxeter transformation of Q′, where y−

C = y+ C when Q′ is

tame. Then y−

C , y+ C are accumulation points in ∆Q, irrational if C is

wild. Are there any other? Yes, y+, y−.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

Irrational accumulation points

For any (Generalized) Kronecker subcategory C = rep(Q′) of rep(Q), denote by y−

C , y+ C the associated eigenvalues of the

Coxeter transformation of Q′, where y−

C = y+ C when Q′ is

tame. Then y−

C , y+ C are accumulation points in ∆Q, irrational if C is

wild. Are there any other? Yes, y+, y−. Theorem The set of accumulation points of ∆Q is the closure of {y−

C , y+ C | C Kronecker subcategory}.

CGMRT 2014, University of Iowa

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Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others

THANK YOU Questions ?

CGMRT 2014, University of Iowa