On projective modules for Frobenius kernels and finite Chevalley groups

Drupieski, Christopher M.

doi:10.1112/blms/bds105

Cited by 3 publications

(6 citation statements)

References 8 publications

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“…In the final section of the paper, we prove the inequality

\begin{equation*} c_{G(\mathbb {F}_{p^r})}(M)\leqslant \frac{r}{r+1}c_{G_{(r)}}(M) \end{equation*}

for an arbitrary finite‐dimensional rational module

M

over a simple algebraic group

G

in good positive characteristic (see Theorem 5.7). This significantly generalizes a result of Lin and Nakano [20, Theorem 3.4(b)] (which considers the case

r=1

) and provides a new proof of the result of Drupieski [8, Theorem 2.3] that

M|_{G({\mathbb {F}}_{p^r})}

is projective if

M|_{G_{(r)}}

is.…”

Section: Introductionsupporting

confidence: 75%

“…In the final section of the paper, we prove the inequality for an arbitrary finite-dimensional rational module 𝑀 over a simple algebraic group 𝐺 in good positive characteristic (see Theorem 5.7). This significantly generalizes a result of Lin and Nakano[20, Theorem 3.4(b)] (which considers the case 𝑟 = 1) and provides a new proof of the result of Drupieski[8, Theorem 2.3] that 𝑀| 𝐺(𝔽 𝑝 𝑟 ) is projective if 𝑀| 𝐺 (𝑟) is.…”

supporting

confidence: 76%

“…(7) (−) 𝑇 denotes the transpose operation. (8) The adjoint action of g ∈ 𝐺 on 𝑥 ∈ 𝔤 is denoted g ⋅ 𝑥 = Ad g(𝑥) = g𝑥g −1 . ( 9)  𝑥 = 𝐺 ⋅ 𝑥 the orbit of an element 𝑥 in 𝔤.…”

Section: Notationmentioning

confidence: 99%

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Commuting varieties and cohomological complexity theory

Lévy

Ngo

Šivic

2022

Journal of London Math Soc

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In this paper we determine, for all r$r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting r$r$‐tuples of nilpotent elements of frakturgln$\mathfrak {gl}_n$. Our main result is that in characteristic ≠2,3$\ne 2,3$, this nilpotent commuting variety has dimension (r+1)⌊n24⌋$(r+1)\lfloor \frac{n^2}{4}\rfloor$ for n⩾4$n\geqslant 4$, r⩾7$r\geqslant 7$. We use this to find the dimension of the (ordinary) r$r$th commuting varieties of frakturgln$\mathfrak {gl}_n$ and fraktursln$\mathfrak {sl}_n$ for the same range of values of r$r$ and n$n$. Our principal motivation is the connection between nilpotent commuting varieties and cohomological complexity of finite group schemes, which we exploit in the last section of the paper to obtain explicit values for complexities of a large family of modules over the r$r$th Frobenius kernel (prefixGLn)(r)$(\operatorname{GL}_n)_{(r)}$. These results indicate an inequality between the complexities of a rational G$G$‐module M$M$ when restricted to G(r)$G_{(r)}$ or to Gfalse(Fprfalse)$G(\mathbb {F}_{p^r})$; we subsequently establish this inequality for every simple algebraic group G$G$ defined over an algebraically closed field of good characteristic, significantly extending the main theorem of Lin and Nakano, Inventiones Mathematicae, 138 (1999), 85–101.

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“…In the final section of the paper, we prove the inequality

\begin{equation*} c_{G(\mathbb {F}_{p^r})}(M)\leqslant \frac{r}{r+1}c_{G_{(r)}}(M) \end{equation*}

for an arbitrary finite‐dimensional rational module

M

over a simple algebraic group

G

in good positive characteristic (see Theorem 5.7). This significantly generalizes a result of Lin and Nakano [20, Theorem 3.4(b)] (which considers the case

r=1

) and provides a new proof of the result of Drupieski [8, Theorem 2.3] that

M|_{G({\mathbb {F}}_{p^r})}

is projective if

M|_{G_{(r)}}

is.…”

Section: Introductionsupporting

confidence: 75%

supporting

confidence: 76%

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Commuting varieties and cohomological complexity theory

Lévy

Ngo

Šivic

2022

Journal of London Math Soc

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“…). Drupieski later generalized this result by proving that, for any

r ⩾ 1

, a finite‐dimensional rational

G

‐module that is projective over

G_{r}

is projective over

G ({double-struckF}_{q})

.…”

Section: Conditions On G For the Existence Of Proper Mock Injective Mmentioning

confidence: 97%

“…Finally, we demonstrate that our constructions (for mock injectives) in this section can be used to show that a formation of the known Parshall Conjecture (cf. ) for infinitely generated modules is false.…”

Section: Introductionmentioning

confidence: 97%

On the existence of mock injective modules for algebraic groups

Hardesty

Nakano

Sobaje

2017

Bull. London Math. Soc.

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Let G be an affine algebraic group scheme over an algebraically closed field k of characteristic p > 0, and let Gr denote the rth Frobenius kernel of G. Motivated by recent work of Friedlander, the authors investigate the class of mock injective G-modules, which are defined to be those rational G-modules that are injective on restriction to Gr for all r 1. In this paper, the authors provide necessary and sufficient conditions for the existence of non-injective mock injective G-modules, thereby answering a question raised by Friedlander. Furthermore, the authors investigate the existence of non-injective mock injectives with simple socles. Interesting cases are discovered that show that this can occur for reductive groups, but will not occur for their Borel subgroups.Thus, every central character is of the form χ = λ + ZJ. If we let π : X(T ) → X(Z J ) denote the quotient map, then π(ZΦ) = ZΦ/ZJ ∼ = ZI where I = Δ\J. It follows that any L J -module M , whose weights all lie in ZΦ, has a central character decomposition of the form M = χ∈ZI M χ . This statement particularly applies to k[U J ]. Lemma 3.3.2. The L J -module k[U J ] has a central character decomposition of the formProof. First, recall that k[U J ] is a polynomial ring given bywhere each x γ has weight γ. For any weight λ ∈ X(T ), the weight space k[U J ] λ is spanned by monomials of the form γ∈Φ + \Φ + J x nγ γ , where λ = γ n γ γ with n γ 0 is any expression for λ. Since there are only finitely many ways to express λ as a non-negative integral combination of elements in Φ + \Φ + J , we can conclude that the weight spaces are finite-dimensional. Fix a central character χ ∈ X(Z J ) which acts non-trivially on k[U J ]. By the remark preceding this lemma and the above description of the weight spaces of k[U J ], it follows that there exists a unique weight λ ∈ NI satisfying χ = λ + ZJ.

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Decomposition of Tensor Products Involving a Steinberg Module

Kildetoft

2017

Algebr Represent Theor

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We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G r or a finite Chevalley group G(F q ). In all three cases, we give formulas reducing this to standard character data for G.Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G r or G(F q ). Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin's tilting conjecture.

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On projective modules for Frobenius kernels and finite Chevalley groups

Cited by 3 publications

References 8 publications

Commuting varieties and cohomological complexity theory

Commuting varieties and cohomological complexity theory

On the existence of mock injective modules for algebraic groups

Decomposition of Tensor Products Involving a Steinberg Module

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