Matrix Problems Associated to Some Brauer Configuration Algebras - PowerPoint PPT Presentation

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Matrix Problems Associated to Some Brauer Configuration Algebras Maurice Auslander Distinguished Lectures Falmouth-USA Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo 04/25/2019 Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Road Map 1 Aims and Scope 2 Brauer Configuration Algebras 3 Some Matrix Problems The Kronecker Problem 4 Helices Helices and Exceptional Sequences Cycles The Four Subspace Problem (FSP) 5 References Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Road Map 1 Aims and Scope 2 Brauer Configuration Algebras 3 Some Matrix Problems The Kronecker Problem 4 Helices Helices and Exceptional Sequences Cycles The Four Subspace Problem (FSP) 5 References Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Road Map 1 Aims and Scope 2 Brauer Configuration Algebras 3 Some Matrix Problems The Kronecker Problem 4 Helices Helices and Exceptional Sequences Cycles The Four Subspace Problem (FSP) 5 References Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Road Map 1 Aims and Scope 2 Brauer Configuration Algebras 3 Some Matrix Problems The Kronecker Problem 4 Helices Helices and Exceptional Sequences Cycles The Four Subspace Problem (FSP) 5 References Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Road Map 1 Aims and Scope 2 Brauer Configuration Algebras 3 Some Matrix Problems The Kronecker Problem 4 Helices Helices and Exceptional Sequences Cycles The Four Subspace Problem (FSP) 5 References Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Aims and Scope Bijections between solutions of the Kronecker problem and the four sub- space problem with indecomposable projective modules over some Brauer configuration algebras are obtained by interpreting elements of some inte- ger sequences as polygons of suitable Brauer configurations. Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Ideas from the Medellin CIMPA School Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Figure: Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Brauer Configuration Algebras Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References Definition Recently, E.L. Green and S. Schroll introduced Brauer configuration algebras as a way to deal with research of algebras of wild repre- sentation type ( Brauer configuration algebras: A generalization of Brauer graph algebras, E.L. Green, S. Schroll, Bull. Sci. Math. vol. 141, 2017, 539-572 ). Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References A Brauer configuration is a tuple Γ = (Γ 0 , Γ 1 , µ, O ) where Γ 0 is a set of vertices, Γ 1 is a set of polygons , µ : Γ 0 → N is a multiplicity function and O is an orientation, such that the following conditions hold: C (1) Every vertex in Γ 0 is a vertex in at least one polygon in Γ 1 . C (2) Every polygon has at least two vertices. C (3) Every polygon has at least a vertex happening more than once (nontruncated vertex). Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References A Brauer configuration is a tuple Γ = (Γ 0 , Γ 1 , µ, O ) where Γ 0 is a set of vertices, Γ 1 is a set of polygons , µ : Γ 0 → N is a multiplicity function and O is an orientation, such that the following conditions hold: C (1) Every vertex in Γ 0 is a vertex in at least one polygon in Γ 1 . C (2) Every polygon has at least two vertices. C (3) Every polygon has at least a vertex happening more than once (nontruncated vertex). Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References A Brauer configuration is a tuple Γ = (Γ 0 , Γ 1 , µ, O ) where Γ 0 is a set of vertices, Γ 1 is a set of polygons , µ : Γ 0 → N is a multiplicity function and O is an orientation, such that the following conditions hold: C (1) Every vertex in Γ 0 is a vertex in at least one polygon in Γ 1 . C (2) Every polygon has at least two vertices. C (3) Every polygon has at least a vertex happening more than once (nontruncated vertex). Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References The cyclic ordering at vertex α is obtained by linearly ordering the list (i.e., V i 1 < · · · < V i t and by adding V i t < V i 1 ). Such a list is said to be the successor sequence at α . Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA

Aims and Scope Brauer Configuration Algebras Some Matrix Problems Helices References The Quiver of a Brauer Configuration Algebra Agust´ ın Moreno Ca˜ nadas jointly with; Pedro Fern´ andez, Jos´ e A. Velez-Marulanda, Hern´ an Giraldo Universidad Nacional de Colombia Matrix Problems Associated to Some Brauer Configuration AlgebrasMaurice Auslander Distinguished LecturesFalmouth-USA