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Daniel K. Nakano

Publications and source records attributed to Daniel K. Nakano.

At least 55 records · Page 3Linked to original sources
arXiv2014

On the structure of cohomology rings of p-nilpotent Lie algebras

In this paper the authors investigate the structure the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose $E_{2}$-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen-Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as ring with the $E_{2}$-term. We present criteria for the collapsing of this spectral sequence and provide many examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.

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arXiv2014

Endotrivial Modules for the General Linear Group in a Nondefining Characteristic

Suppose that $G$ is a finite group such that $\operatorname{SL}(n,q)\subseteq G \subseteq \operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\operatorname{GL}(n,q)$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.

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arXiv2013

Extensions for Finite Chevalley Groups III: Rational and Generic Cohomology

Let $G$ be a connected reductive algebraic group and $B$ be a Borel subgroup defined over an algebraically closed field of characteristic $p>0$. In this paper, the authors study the existence of generic $G$-cohomology and its stability with rational $G$-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of $G$ and $B$-cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of $G({\mathbb F}_{q})$ are established which generalizes earlier work of Hiller, in addition to stability ranges for generic cohomology which improves on seminal work of Cline, Parshall, Scott and van der Kallen.

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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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arXiv2012

Bounding extensions for finite groups and Frobenius kernels

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ:G->G be a strict endomorphism (i. e., the subgroup G(σ) of σ-fixed points is finite). Also, let G_σ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the degree m cohomology H^m(G(σ),L) of any irreducible kG(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. A similar result holds for the degree m cohomology of G_σ. These bounds are actually established for the degree m extension groups Ext^m_{G(σ)}(L,L') between irreducible kG(σ)-modules L and L', with again a similar result holding for G_σ. In these Ext^m results, of interest in their own right, the bounds depend also on L, or, more precisely, on length of the p-adic expansion of the highest weight associated to L. All bounds are, nevertheless, independent of the characteristic p. These results extend earlier work of Parshall and Scott for rational representations of algebraic groups G. We also show that one can find bounds independent of the prime for the Cartan invariants of G(σ) and G_σ, and even for the lengths of the underlying PIMs. These bounds, which depend only on the root system of G and the "height" of σ, provide in a strong way an affirmative answer to a question of Hiss, for the special case of finite groups G(σ) of Lie type in the defining characteristic.

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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

On the vanishing ranges for the cohomology of finite groups of Lie type II

The computation of the cohomology for finite groups of Lie type in the describing characteristic is a challenging and difficult problem. In earlier work, the authors constructed an induction functor which takes modules over the finite group of Lie type to modules for the ambient algebraic group G. In particular this functor when applied to the trivial module yields a natural G-filtration. This filtration was utilized in the earlier work to determine the first non-trivial cohomology class when the underlying root system is of type A_{n} or C_{n}. In this paper the authors extend these results toward locating the first non-trivial cohomology classes for the remaining finite groups of Lie type (i.e., the underlying root system is of type B_{n}, C_{n}, D_{n}, E_{6}, E_{7}, E_{8}, F_{4}, and G_{2}) when the prime is larger than the Coxeter number.

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arXiv2011

On the support varieties of Demazure modules

We consider the support varieties of Demazure modules, certain $B$-modules important in the representation theory of reductive groups. In many cases we are able to compute these support varieties over $B_1$, the first Frobenius kernel of a Borel subgroup, and relate them to orbital varieties. We provide a complete calculation of the support varieties when the underlying algebraic group has type $A_1$ or $A_2$. We also consider the saturation ($G$ orbit of) $B_1$ support varieties for Demazure modules and show that the inclusion relations among these varieties are compatible with the Bruhat order.

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arXiv2011

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group $G$, a parabolic subgroup $P_J$, and its unipotent radical $U_J$, we determine the ring structure of the cohomology ring $H^\bullet((U_J)_1,k)$. We also obtain new results on computing $H^\bullet((P_J)_1,L(λ))$ as an $L_J$-module where $L(λ)$ is a simple $G$-module with high weight $λ$ in the closure of the bottom $p$-alcove. Finally, we provide generalizations of all our results to the quantum situation.

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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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arXiv2011

Endotrivial modules for finite groups schemes II

It is well known that if G is a finite group then the group of endotrivial modules is finitely generated. In this paper we prove that for an arbitrary finite group scheme G, and for any fixed integer n > 0, there are only finitely many isomorphism classes of endotrivial modules of dimension n. This provides evidence to support the speculation that the group of endotrivial modules for a finite group scheme is always finitely generated. The result also has some applications to questions about lifting and twisting the structure of endotrivial modules in the case that G is an infinitesimal group scheme associated to an algebraic group.

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arXiv2011

Cohomology for quantum groups via the geometry of the nullcone

Let $ζ$ be a complex $\ell$th root of unity for an odd integer $\ell>1$. For any complex simple Lie algebra $\mathfrak g$, let $u_ζ=u_ζ({\mathfrak g})$ be the associated "small" quantum enveloping algebra. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when $l$ (resp., $p$) is smaller than the Coxeter number $h$ of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible $G$-modules stipulates that $p \geq h$. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra $\opH^\bullet(u_ζ,{\mathbb C})$ of the small quantum group. When $\ell>h$, this cohomology algebra has been calculated by Ginzburg and Kumar \cite{GK}. Our result requires powerful tools from complex geometry and a detailed knowledge of the geometry of the nullcone of $\mathfrak g$. In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra $u$ in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if $M$ is a finite dimensional $u_ζ$-module, then $\opH^\bullet(u_ζ,M)$ is a finitely generated $\opH^\bullet(u_ζ,\mathbb C)$-module, and we obtain new results on the theory of support varieties for $u_ζ$.

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arXiv2010

Detecting Cohomology for Lie Superalgebras

In this paper we use invariant theory to develop the notion of cohomological detection for Type I classical Lie superalgebras. In particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras. These results are used later to affirmatively answer questions, which were originally posed in \cite{BKN1} and \cite{BaKN}, about realizing support varieties for Lie superalgebras via rank varieties constructed for the smaller detecting subalgebras.

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arXiv2010

On the vanishing ranges for the cohomology of finite groups of Lie type

Let $G({\mathbb F}_{q})$ be a finite Chevalley group defined over the field of $q=p^{r}$ elements, and $k$ be an algebraically closed field of characteristic $p>0$. A fundamental open and elusive problem has been the computation of the cohomology ring $\opH^{\bullet}(G({\mathbb F}_{q}),k)$. In this paper we determine initial vanishing ranges which improves upon known results. For root systems of type $A_n$ and $C_n$, the first non-trivial cohomology classes are determined when $p$ is larger than the Coxeter number (larger than twice the Coxeter number for type $A_n$ with $n>1$ and $r >1$). In the process we make use of techniques involving line bundle cohomology for the flag variety $G/B$ and its relation to combinatorial data from Kostant Partition Functions.

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arXiv2010

Differentiating the Weyl generic dimension formula and support varieties for quantum groups

The authors compute the support varieties of all irreducible modules for the small quantum group $u_ζ(\mathfrak{g})$, where $\mathfrak{g}$ is a simple complex Lie algebra, and $ζ$ is a primitive $\ell$-th root of unity with $\ell$ larger than the Coxeter number of $\mathfrak{g}$. The calculation employs the prior calculations and techniques of Ostrik and of Nakano--Parshall--Vella, as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan-Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel $G_1$ of a reductive algebraic group scheme $G$ defined over the prime field $\mathbb{F}_p$.

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arXiv2009

On the cohomology of Young modules for the symmetric group

The main result of this paper is an application of the topology of the space $Q(X)$ to obtain results for the cohomology of the symmetric group on $d$ letters, $Σ_d$, with `twisted' coefficients in various choices of Young modules and to show that these computations reduce to certain natural questions in representation theory. The authors extend classical methods for analyzing the homology of certain spaces $Q(X)$ with mod-$p$ coefficients to describe the homology $\HH_{\bullet}(Σ_d, V^{\otimes d})$ as a module for the general linear group $GL(V)$ over an algebraically closed field $k$ of characteristic $p$. As a direct application, these results provide a method of reducing the computation of $\text{Ext}^{\bullet}_{Σ_{d}}(Y^λ,Y^μ)$ (where $Y^λ$, $Y^μ$ are Young modules) to a representation theoretic problem involving the determination of tensor products and decomposition numbers. In particular, in characteristic two, for many $d$, a complete determination of $\Hs Y^λ)$ can be found. This is the first nontrivial class of symmetric group modules where a complete description of the cohomology in all degrees can be given. For arbitrary $d$ the authors determine $\HH^i(Σ_d,Y^λ)$ for $i=0,1,2$. An interesting phenomenon is uncovered--namely a stability result reminiscent of generic cohomology for algebraic groups. For each $i$ the cohomology $\HH^i(Σ_{p^ad}, Y^{p^aλ})$ stabilizes as $a$ increases. The methods in this paper are also powerful enough to determine, for any $p$ and $λ$, precisely when $\HH^{\bullet}(\sd,Y^λ)=0$. Such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus.

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