Dissertation Defense

Chloé Napier -- Dissertation Defense

Dissertation Defense

Speaker:  Chloé Napier, University of Kentucky

Extensions between modules defined by lattice paths in the preprojective algebra
 
In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. 
 
The main goal is to find a combinatorial description for when certain cluster variables are compatible or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc and Schröer that answer this question in the case of the Grassmannian by using various methods. We introduce the notion of extending a module, describe how add or remove operators applied to pairs of modules affects the extension space between them and provide homological and combinatorial conditions that determine when two arbitrary modules have trivial extension.
Date:
-
Location:
745 POT

Chloé Napier -- Dissertation Defense

Dissertation Defense

Speaker:  Chloé Napier, University of Kentucky

Extensions between modules defined by lattice paths in the preprojective algebra
 
In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. 
 
The main goal is to find a combinatorial description for when certain cluster variables are compatible or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc and Schröer that answer this question in the case of the Grassmannian by using various methods. We introduce the notion of extending a module, describe how add or remove operators applied to pairs of modules affects the extension space between them and provide homological and combinatorial conditions that determine when two arbitrary modules have trivial extension.
Date:
-
Location:
745 POT

Williem Rizer - Doctoral Defense

Doctoral Defense (Note time)

Speaker:  Williem Rizer

Title:  Combinatorial models for nonnegativity in flag varieties

Abstract:  

The nonnegative Grassmannian admits a widely studied cell decomposition due to Alexander Postnikov, whose cells are indexed by positroids and modeled by several equivalent combinatorial objects. Subsequent work by authors including Lauren Williams, Suho Oh and Carolina Benedetti has further developed the combinatorics and geometry of these structures. 

In this talk, we will extend some of Postnikov’s combinatorial framework to the nonnegative flag variety. We introduce flag positroid pipe dreams, a diagrammatic model analogous to Le diagrams, together with associated directed graphs and networks that parameterize Richardson cells indexed by flag positroids. Using this framework, we give a constructive proof of a conjecture from Benedetti-Chavez-Tamayo in the case of nonnegatively representable quotients, giving a complete characterization of all positroids of which a fixed positroid is such a quotient in terms of decorated permutations and diagrammatic data. 

Our approach also highlights the connection between flag positroids and intervals in the Bruhat order. Building on work of onathan Boretsky, Christopher Eur and Williams, we show that flag positroid pipe dreams are in bijection with Bruhat intervals, where the number of ones in the diagram encodes the interval length and hence the dimension of the corresponding Richardson cell. Together, these results provide a unified combinatorial perspective on nonnegativity in flag varieties.

 

Date:
-
Location:
POT 745

Williem Rizer - Doctoral Defense

Doctoral Defense (Note time)

Speaker:  Williem Rizer

Title:  Combinatorial models for nonnegativity in flag varieties

Abstract:  

The nonnegative Grassmannian admits a widely studied cell decomposition due to Alexander Postnikov, whose cells are indexed by positroids and modeled by several equivalent combinatorial objects. Subsequent work by authors including Lauren Williams, Suho Oh and Carolina Benedetti has further developed the combinatorics and geometry of these structures. 

In this talk, we will extend some of Postnikov’s combinatorial framework to the nonnegative flag variety. We introduce flag positroid pipe dreams, a diagrammatic model analogous to Le diagrams, together with associated directed graphs and networks that parameterize Richardson cells indexed by flag positroids. Using this framework, we give a constructive proof of a conjecture from Benedetti-Chavez-Tamayo in the case of nonnegatively representable quotients, giving a complete characterization of all positroids of which a fixed positroid is such a quotient in terms of decorated permutations and diagrammatic data. 

Our approach also highlights the connection between flag positroids and intervals in the Bruhat order. Building on work of onathan Boretsky, Christopher Eur and Williams, we show that flag positroid pipe dreams are in bijection with Bruhat intervals, where the number of ones in the diagram encodes the interval length and hence the dimension of the corresponding Richardson cell. Together, these results provide a unified combinatorial perspective on nonnegativity in flag varieties.

 

Date:
-
Location:
POT 745