Determinants, Pfaffians, symmetric quivers, and symmetric varieties

Type A quiver loci are a class of generalized determinantal varieties. Special cases include classical determinantal varieties and varieties of complexes. Since the 1980s, mathematicians have found connections between these quiver loci and Schubert varieties in type A flag varieties. These connections were used to gain insights into commutative-algebraic properties of type A quiver loci (e.g., singularities, Hilbert series).  

In this talk, I will motivate and recall some of this story. I will then discuss the related setting of H. Derksen and J. Weyman's symmetric quivers and their representation varieties. Special cases include varieties defined by minors (or Pfaffians) of symmetric and skew-symmetric matrices. I will show how one can unify the study of commutative-algebraic properties of finite type symmetric quiver representation varieties with corresponding properties for Borel orbit closures in symmetric varieties G/K (G = general linear group, K = orthogonal or symplectic subgroup). Finally, I will provide some commutative-algebraic consequences.

This is joint work with Ryan Kinser and Martina Lanini. 

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