Alex Weekes
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View article: Fundamental Monopole Operators and Embeddings of Kac-Moody Affine Grassmannian Slices
Fundamental Monopole Operators and Embeddings of Kac-Moody Affine Grassmannian Slices Open
Braverman, Finkelberg, and Nakajima define Kac-Moody affine Grassmannian slices as Coulomb branches of $3d$ ${\mathcal{N}}=4$ quiver gauge theories and prove that their Coulomb branch construction agrees with the usual loop group definitio…
View article: Braid group actions, Baxter polynomials, and affine quantum groups
Braid group actions, Baxter polynomials, and affine quantum groups Open
It is a classical result in representation theory that the braid group $\mathscr{B}_\mathfrak{g}$ of a simple Lie algebra $\mathfrak{g}$ acts on any integrable representation of $\mathfrak{g}$ via triple products of exponentials in its Che…
View article: Lie algebra actions on module categories for truncated shifted yangians
Lie algebra actions on module categories for truncated shifted yangians Open
We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction fun…
View article: Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices
Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices Open
Braverman, Finkelberg, and Nakajima define Kac-Moody affine Grassmannian slices as Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and prove that their Coulomb branch construction agrees with the usual loop group definition …
View article: Lie algebra actions on module categories for truncated shifted Yangians
Lie algebra actions on module categories for truncated shifted Yangians Open
We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's induction and restriction fun…
View article: Coulomb branches of quiver gauge theories with symmetrizers
Coulomb branches of quiver gauge theories with symmetrizers Open
We generalize the mathematical definition of Coulomb branches of 3 -dimensional \mathcal N= 4 SUSY quiver gauge theories due to Nakajima (2016) and Braverman et al. (2018, 2019) to the cases with symmetrizers . We obtain generalized affine…
View article: On a conjecture of Pappas and Rapoport about the standard local model for GL_<sub> <i>d</i> </sub>
On a conjecture of Pappas and Rapoport about the standard local model for GL_<sub> <i>d</i> </sub> Open
In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of n × n {n\times n} matrices. We give a positive answer to their …
View article: Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians
Hamiltonian reduction for affine Grassmannian slices and truncated shifted Yangians Open
Generalized affine Grassmannian slices provide geometric realizations for weight spaces of representations of semisimple Lie algebras. They are also Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper, we prove th…
View article: Crystals and monodromy of Bethe vectors
Crystals and monodromy of Bethe vectors Open
Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras\nacting on tensor product multiplicity spaces for g-representations. These\nalgebras depend on a parameter which is a point in the Deligne-Mumford moduli\nspace of ma…
View article: The Equations Defining Affine Grassmannians in Type A and a Conjecture of Kreiman, Lakshmibai, Magyar, and Weyman
The Equations Defining Affine Grassmannians in Type A and a Conjecture of Kreiman, Lakshmibai, Magyar, and Weyman Open
The affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of…
View article: A quantum Mirković-Vybornov isomorphism
A quantum Mirković-Vybornov isomorphism Open
We present a quantization of an isomorphism of Mirković and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in to a slice between spherical Schubert varieties in the affine Grassmannian of (with w…
View article: On a conjecture of Pappas and Rapoport about the standard local model\n for $GL_d$
On a conjecture of Pappas and Rapoport about the standard local model\n for $GL_d$ Open
In their study of local models of Shimura varieties for totally ramified\nextensions, Pappas and Rapoport posed a conjecture about the reducedness of a\ncertain subscheme of $n \\times n$ matrices. We give a positive answer to their\nconje…
View article: On a conjecture of Pappas and Rapoport about the standard local model for $GL_d$
On a conjecture of Pappas and Rapoport about the standard local model for $GL_d$ Open
In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of $n \times n$ matrices. We give a positive answer to their conjectur…
View article: Highest weights for truncated shifted Yangians and product monomial crystals
Highest weights for truncated shifted Yangians and product monomial crystals Open
Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian.We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest …
View article: Coulomb branches of quiver gauge theories with symmetrizers
Coulomb branches of quiver gauge theories with symmetrizers Open
We generalize the mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ SUSY quiver gauge theories in arXiv:1503.03676, arXiv:1601.03686, arXiv:1604.03625 to the cases with symmetrizers. We obtain generalized affine…
View article: Generators for Coulomb branches of quiver gauge theories
Generators for Coulomb branches of quiver gauge theories Open
We study the Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories, focusing on the generators for their quantized coordinate rings. We show that there is a surjective map from a shifted Yangian onto the quantized Coulomb branch, …
View article: Symplectic leaves for generalized affine Grassmannian slices
Symplectic leaves for generalized affine Grassmannian slices Open
The generalized affine Grassmannian slices $\overline{\mathcal{W}}_μ^λ$ are algebraic varieties introduced by Braverman, Finkelberg, and Nakajima in their study of Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories. We prove a …
View article: On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products
On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products Open
Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights $λ$ and $μ$ for a Lie algebra $\mathfrak{g}$, which we will assume is simply-l…
View article: The equations defining affine Grassmannians in type A and a conjecture\n of Kreiman, Lakshmibai, Magyar, and Weyman
The equations defining affine Grassmannians in type A and a conjecture\n of Kreiman, Lakshmibai, Magyar, and Weyman Open
The affine Grassmannian of $SL_n$ admits an embedding into the Sato\nGrassmannian, which further admits a Pl\\"ucker embedding into the\nprojectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman\ndescribe the linear p…
View article: Reducedness of affine Grassmannian slices in type A
Reducedness of affine Grassmannian slices in type A Open
We prove in type A a conjecture which describes the ideal of transversal slices to spherical Schubert varieties in the affine Grassmannian. As a corollary, we prove a modular description (due to Finkelberg-Mirković) of the spherical Schube…
View article: A quantum Mirković-Vybornov isomorphism
A quantum Mirković-Vybornov isomorphism Open
We present a quantization of an isomorphism of Mirković and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in $\mathfrak{gl}_N$ , to a slice between spherical Schubert varieties in the affine Grass…
View article: Highest Weights for Truncated Shifted Yangians
Highest Weights for Truncated Shifted Yangians Open
Truncated shifted Yangians are a family of algebras which are conjectured to quantize slices to Schubert varieties in the affine Grassmannian. In this thesis we study the highest weight theory of these algebras, and explore connections wit…
View article: Highest weights for truncated shifted Yangians and product monomial\n crystals
Highest weights for truncated shifted Yangians and product monomial\n crystals Open
Truncated shifted Yangians are a family of algebras which are natural\nquantizations of slices in the affine Grassmannian. We study the highest weight\nrepresentations of these algebras. In particular, we conjecture that the\npossible high…