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Gus Lonergan

Publications and source records attributed to Gus Lonergan.

7 recordsLinked to original sources
arXiv2021

Parity Sheaves and Smith Theory

Let $p$ be a prime number and let $X$ be a complex algebraic variety with an action of $\mathbb{Z}/p\mathbb{Z}$. We develop the theory of parity complexes in a certain $2$-periodic localization of the equivariant constructible derived category $D^b_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_p)$. Under certain assumptions, we use this to define a functor from the category of parity sheaves on $X$ to the category of parity sheaves on the fixed-point locus $X^{\mathbb{Z}/p\mathbb{Z}}$. This may be thought of as a categorification of Smith theory. When $X$ is the affine Grassmannian associated to some complex reductive group, our functor gives a geometric construction of the Frobenius-contraction functor recently defined by M. Gros and M. Kaneda via the geometric Satake equivalence.

math.RT↗
arXiv2018

Steenrod Operators, the Coulomb Branch and the Frobenius Twist, I

In Part I, we use Steenrod's construction to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We will also demonstrate the corresponding result for the $K$-theoretic version of the quantum Coulomb branch. In Part II, we use the same method to construct a functor of categorical $p$-center between the derived Satake categories with and without loop-rotation, which extends the Frobenius twist functor for representations of the dual group.

math.RT↗
arXiv2017

The Lower Central Series of the Quotient of a Free Algebra

Let $L_i(R)$ denote the $i^{\text{th}}$ term of the lower central series of an associative algebra $R$, and let $B_i(R)=L_i(R)/L_{i+1}(R)$. We show that $B_2(\mathbb{C} / P)\cong Ω^2((\mathbb{C} / P)_{ab})$, for all homogeneous or quasihomogeneous $P$ with square-free abelianization. Our approach generalizes that of Balagovic and Balasubramanian in 2010, which in turn developed from that of Dobrovolska, Kim, and Ma in 2007. We also use ideas of Feign and Shoikhet in 2006, who initiated the study of the groups $B_i(R)$.

math.RA↗
arXiv2017

Signatures of Multiplicity Spaces in tensor products of $\mathfrak{sl}_2$ and $U_q(\mathfrak{sl}_2)$ Representations

We study multiplicity space signatures in tensor products of representations of $\mathfrak{sl}_2$ and $U_q(\mathfrak{sl}_2)$, and give some applications. We completely classify definite multiplicity spaces for generic tensor products of $\mathfrak{sl}_2$ Verma modules. This provides a classification of a family of unitary representations of a basic quantized quiver variety, one of the first such classifications for any quantized quiver variety. We use multiplicity space signatures to provide the first real critical point lower bound for generic $\mathfrak{sl}_2$ master functions. As a corollary of this bound, we obtain a simple and asymptotically correct approximation for the number of real critical points of a generic $\mathfrak{sl}_2$ master function. We obtain a formula for multiplicity space signatures in tensor products of finite dimensional simple $U_q(\mathfrak{sl}_2)$ representations. Our formula also gives multiplicity space signatures in generic tensor products of $\mathfrak{sl}_2$ Verma modules and generic tensor products of real $U_q(\mathfrak{sl}_2)$ Verma modules. Our results have relations with knot theory, statistical mechanics, quantum physics, and geometric representation theory.

math.RT↗
arXiv2017

A remark on descent for Coxeter groups

Let $Γ$ be a finite Coxeter group with reflection representation $R$. We show that a $Γ$-equivariant quasicoherent sheaf on $R$ descends to the quotient space $R//Γ$ if it descends to the quotient space $R//\langle s_i\rangle$ for every simple reflection $s_i\in Γ$.

math.RT↗
arXiv2016

A Strong Splitting of the Frobenius Morphism on the Algebra of Distributions of $SL_2$

Let $p$ be a prime number, and let $Dist(SL_2)$ be the algebra of distributions, supported at $1$, on the algebraic group $SL_2$ over $\mathbb{F}_p$. The Frobenius map $Fr:SL_2\to SL_2$ induces a map $Fr:Dist(SL_2)\to Dist(SL_2)$ which is in particular a surjective algebra homomorphism. In this note, we construct a section of this map, whenever $p\geq 3$. The main ingredient of this construction is a certain congruence modulo $p^3$, reminiscent of the congruence $\binom{np}{p}\equiv n\mod p^3$.

math.RT↗