arXiv subjects

Brian D. Boe

Publications and source records attributed to Brian D. Boe.

16 recordsLinked to original sources

arXiv2024

Singular loci of Schubert varieties and the Lookup Conjecture in type $\tilde A_{2}$

We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$. Together with the results of Graham and Li for spiral elements, this allows us to explicitly identify the maximal singular and nrs points in any Schubert variety of type $\tilde{A}_2$. Comparable results are not known for any other infinite-dimensional Kac-Moody flag variety (except for type $\tilde{A}_1$, where every Schubert variety is rationally smooth). As a consequence, we deduce that if $x$ is a point in a non-spiral Schubert variety $X_w$, then $x$ is nrs in $X_w$ if and only if there are more than $\dim X_w$ curves in $X_w$ through $x$ which are stable under the action of a maximal torus, as is true for Schubert varieties in (finite) type $A$. Combined with the work of Graham and Li for spiral Schubert varieties, this implies the Lookup Conjecture for $\tilde{A}_2$.

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arXiv2020

Complexity and Support Varieties for Type P Lie Superalgebras

We compute the complexity, z-complexity, and support varieties of the (thick) Kac modules for the Lie superalgebras of type P. We also show the complexity and the z-complexity have geometric interpretations in terms of support and associated varieties; these results are in agreement with formulas previously discovered for other classes of Lie superalgebras. Our main technical tool is a recursive algorithm for constructing projective resolutions for the Kac modules. The indecomposable projective summands which appear in a given degree of the resolution are explicitly described using the combinatorics of weight diagrams. Surprisingly, the number of indecomposable summands in each degree can be computed exactly: we give an explicit formula for the corresponding generating function.

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arXiv2019

Tensor Triangular Geometry for Quantum Groups

Let $\mathfrak g$ be a complex simple Lie algebra and let $U_ζ({\mathfrak g})$ be the corresponding Lusztig ${\mathbb Z}[q,q^{-1}]$-form of the quantized enveloping algebra specialized to an $\ell$th root of unity. Moreover, let $\mod(U_ζ({\mathfrak g}))$ be the braided monoidal category of finite-dimensional modules for $U_ζ({\mathfrak g})$. In this paper we classify the thick tensor ideals of $\mod(U_ζ({\mathfrak g}))$ and compute the prime spectrum of the stable module category associated to $\text{mod}(U_ζ({\mathfrak g}))$ as defined by Balmer.

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arXiv2018

Tensor Triangular Geometry for Classical Lie Superalgebras

Tensor triangular geometry as introduced by Balmer is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category. In this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the Balmer spectrum. For a classical Lie superalgebra ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$, we construct a Zariski space from a detecting subalgebra of ${\mathfrak g}$ and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional ${\mathfrak g}$-modules which are semisimple over ${\mathfrak g}_{\bar{0}}$.

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arXiv2015

Extensions for Generalized Current Algebras

Given a complex semisimple Lie algebra ${\mathfrak g}$ and a commutative ${\mathbb C}$-algebra $A$, let ${\mathfrak g}[A] = {\mathfrak g} \otimes A$ be the corresponding generalized current algebra. In this paper we explore questions involving the computation and finite-dimensionality of extension groups for finite-dimensional ${\mathfrak g}[A]$-modules. Formulas for computing $\operatorname{Ext}^{1}$ and $\operatorname{Ext}^{2}$ between simple ${\mathfrak g}[A]$-modules are presented. As an application of these methods and of the use of the first cyclic homology, we completely describe $\operatorname{Ext}^{2}_{{\mathfrak g}[t]}(L_{1},L_{2})$ for ${\mathfrak g}=\mathfrak{sl}_{2}$ when $L_{1}$ and $L_{2}$ are simple ${\mathfrak g}[t]$-modules that are each given by the tensor product of two evaluation modules.

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arXiv2013

Bounding the dimensions of rational cohomology groups

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple simply-connected algebraic group over $k$ that is defined and split over the prime field $\mathbb{F}_p$. In this paper we investigate situations where the dimension of a rational cohomology group for $G$ can be bounded by a constant times the dimension of the coefficient module. We then demonstrate how our results can be applied to obtain effective bounds on the first cohomology of the symmetric group. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.

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arXiv2011

Second cohomology for finite groups of Lie type

Let $G$ be a simple, simply-connected algebraic group defined over $\mathbb{F}_p$. Given a power $q = p^r$ of $p$, let $G(\mathbb{F}_q) \subset G$ be the subgroup of $\mathbb{F}_q$-rational points. Let $L(λ)$ be the simple rational $G$-module of highest weight $λ$. In this paper we establish sufficient criteria for the restriction map in second cohomology $H^2(G,L(λ)) \rightarrow H^2(G(\mathbb{F}_q),L(λ))$ to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on $p$ and $q$ provided $λ$ is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe $H^2(G(\mathbb{F}_q),L(λ))$ in terms of rational cohomology for $G$. We apply our techniques to compute $H^2(G(\mathbb{F}_q),L(λ))$ in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.

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arXiv2011

First cohomology for finite groups of Lie type: simple modules with small dominant weights

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $H^1(G(\mathbb{F}_q),L(λ))$ where $L(λ)$ is the simple $G$-module of highest weight $λ$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $λ$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $λ$ is a minimal nonzero dominant weight.

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arXiv2011

Complexity for Modules Over the Classical Lie Superalgebra gl(m|n)

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus \mathfrak{g}_{\bar{1}}$ be a classical Lie superalgebra and $\mathcal{F}$ be the category of finite dimensional $\mathfrak{g}$-supermodules which are completely reducible over the reductive Lie algebra $\mathfrak{g}_{\bar{0}}$. In an earlier paper the authors demonstrated that for any module $M$ in $\mathcal{F}$ the rate of growth of the minimal projective resolution (i.e., the complexity of $M$) is bounded by the dimension of $\mathfrak{g}_{\bar{1}}$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\mathfrak{gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.

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arXiv2009

Complexity and module varieties for classical Lie superalgebras

Let g=g_{0} \oplus g_{1} be a classical Lie superalgebra and F be the category of finite dimensional g-supermodules which are semisimple over g_{0}. In this paper we investigate the homological properties of the category F. In particular we prove that F is self-injective in the sense that all projective supermodules are injective. We also show that all supermodules in F admit a projective resolution with polynomial rate of growth and, hence, one can study complexity in F. If g is a Type I Lie superalgebra we introduce support varieties which detect projectivity and are related to the associated varieties of Duflo and Serganova. If in addition g has a (strong) duality then we prove that the conditions of being tilting or projective are equivalent.

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arXiv2008

Cohomology of quantum groups: An analog of Kostant's Theorem

We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostant's cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $\mathfrak{g} = \mathfrak{sl}(n)$. We also show that Kostant's formula holds when $q$ is specialized to an $\ell$-th root of unity for odd $\ell \ge h-1$ (where $h$ is the Coxeter number of $\mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.

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arXiv2008

Ext^1-quivers for the Witt algebra W(1,1)

Let g be the finite dimensional Witt algebra W(1,1) over an algebraically closed field of characteristic p > 3. It is well known that all simple W(1,1)-modules are finite dimensional. Each simple module admits a character χin g^*. Given such a χone can form the (finite dimensional) reduced enveloping algebra u(g,χ). The simple modules for u(g,χ) are precisely those simple W(1,1)-modules admitting the character χ. In this paper the authors compute Ext^1 between pairs of simple modules for u(g,χ).

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arXiv2008

Category $\mathcal O$ for the Virasoro algebra: Cohomology and Koszulity

We investigate blocks of the Category $\mathcal O$ for the Virasoro algebra over the complex numbers. We demonstrate that the blocks have Kazhdan-Lusztig theories, and that the truncated blocks give rise to interesting Koszul algebras. The simple modules have BGG resolutions, and from this we compute the extensions between Verma modules and simple modules, and between pairs of simple modules.

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arXiv2007

Cohomology and Support Varieties for Lie Superalgebras II

In \cite{BKN} the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for $\mathfrak{gl}(m|n)$. The latter result verifies our earlier conjecture for $\mathfrak{gl}(m|n)$. In our investigation we also delineate several of the major differences between Type I versus Type II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.

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arXiv2006

Kostant modules in blocks of category ${\mathcal O}_S$

In this paper the authors investigate infinite-dimensional representations $L$ in blocks of the relative (parabolic) category ${\mathcal O}_S$ for a complex simple Lie algebra, having the property that the cohomology of the nilradical with coefficients in $L$ ``looks like'' the cohomology with coefficients in a finite-dimensional module, as in Kostant's theorem. A complete classification of these ``Kostant modules'' in regular blocks for maximal parabolics in the simply laced types is given. A complete classification is also given in arbitrary (singular) blocks for Hermitian symmetric categories.

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arXiv2006

Cohomology and Support Varieties for Lie Superalgebras

We study cohomology for classical Lie superalgebras $\mathfrak{g}$ (e.g. gl(m|n)) over the complex numbers. Using results from invariant theory, we show that there exist subsuperalgebras which detect the cohomology of $\mathfrak{g}.$ Furthermore, drawing inspiration from finite groups and restricted Lie algebras, we introduce the notion of support varieties in this context. One can prove these varieties have many of the desirable properties of such a theory. They also appear to provide new interpretations for certain combinatorics associated with Lie superalgebras.

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