Accumulation points of real Schur roots Charles Paquette November 22 - PowerPoint PPT Presentation

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 2014, University of Iowa

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

Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others Settings k = ¯ k is an algebraically closed field. Q = ( Q 0 , Q 1 ) is a connected acyclic quiver with Q 0 = { 1 , 2 , . . . , n } . CGMRT 2014, University of Iowa

Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others Settings k = ¯ k is an algebraically closed field. Q = ( Q 0 , Q 1 ) is a connected acyclic quiver with Q 0 = { 1 , 2 , . . . , n } . We may choose an admissible ordering, that is, j → i ∈ Q 1 implies i < j . CGMRT 2014, University of Iowa

Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others Settings k = ¯ k is an algebraically closed field. Q = ( Q 0 , Q 1 ) is a connected acyclic quiver with Q 0 = { 1 , 2 , . . . , n } . We may choose an admissible ordering, that is, j → i ∈ Q 1 implies i < j . rep ( Q ) denotes the category of finite dimensional representations of Q over k . CGMRT 2014, University of Iowa

Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others Settings k = ¯ k is an algebraically closed field. Q = ( Q 0 , Q 1 ) is a connected acyclic quiver with Q 0 = { 1 , 2 , . . . , n } . We may choose an admissible ordering, that is, j → i ∈ Q 1 implies i < j . rep ( Q ) denotes the category of finite dimensional representations of Q over k . Given M ∈ rep ( Q ), we denote by d M ∈ Z n ≥ 0 its dimension vector. CGMRT 2014, University of Iowa

Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others Settings k = ¯ k is an algebraically closed field. Q = ( Q 0 , Q 1 ) is a connected acyclic quiver with Q 0 = { 1 , 2 , . . . , n } . We may choose an admissible ordering, that is, j → i ∈ Q 1 implies i < j . rep ( Q ) denotes the category of finite dimensional representations of Q over k . Given M ∈ rep ( Q ), we denote by d M ∈ Z n ≥ 0 its dimension vector. We denote by �− , −� the Euler-Ringel form of Q , that is, � d M , d N � = dim k Hom ( M , N ) − dim k Ext 1 ( M , N ) . CGMRT 2014, University of Iowa

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

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 d M is a Schur root. CGMRT 2014, University of Iowa

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 d M is a Schur root. We then call d M  real , if � d M , d M � = 1;   imaginary , if � d M , d M � ≤ 0;  isotropic , if � d M , d M � = 0;   strictly imaginary , if � d M , d M � < 0;  CGMRT 2014, University of Iowa

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 R n in the positive orthant. CGMRT 2014, University of Iowa

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 R n in the positive orthant. We denote by [ d ] the ray of d ∈ Z n ≥ 0 . CGMRT 2014, University of Iowa

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 R n in the positive orthant. We denote by [ d ] the ray of d ∈ Z n ≥ 0 . A Schur root that is real or isotropic is uniquely determined by its ray. CGMRT 2014, University of Iowa

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 R n in the positive orthant. We denote by [ d ] the ray of d ∈ Z n ≥ 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

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

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

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

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 ˜ A 2 , 1 CGMRT 2014, University of Iowa

Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others Example 1 CGMRT 2014, University of Iowa

�� � 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 CGMRT 2014, University of Iowa

Introduction Accumulation points - Examples Accumulation points - Canonical decomposition Accumulation points - Rational ones Accumulation points - Others Example 2 CGMRT 2014, University of Iowa

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 d 1 , . . . , d r such that CGMRT 2014, University of Iowa

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 d 1 , . . . , d r such that ext 1 ( d i , d j ) = 0 whenever i � = j. CGMRT 2014, University of Iowa

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 d 1 , . . . , d r such that ext 1 ( d i , d j ) = 0 whenever i � = j. the coefficient of a strictly imaginary Schur root is one. CGMRT 2014, University of Iowa

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 d 1 , . . . , d r such that ext 1 ( d i , d j ) = 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