Zero-separating invariants for finite groups

We fix a field k of characteristic p. For a finite group G denote by δ(G) and σ(G) respectively the minimal number d, such that for every finite dimensional representation V of G over k and every v ∈ V^G \ {0} or v ∈ V \ {0} respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) = 0. Let P be a Sylow-p-subgroup of G (which we take to be trivial if the group order is not divisble by p). We show that δ(G) = |P|. If N_G(P)/P is cyclic, we show σ(G) ≥ |N_G(P)|. If G is p-nilpotent and P is not normal in G, we show σ(G) ≤ |G|/l , where l is the smallest prime divisor of |G|. These results extend known results in the non-modular case to the modular case

Separating Invariants for the Basic G_a actions

Abstract. We explicitly construct a finite set of separating invariants for the basic G_a -actions. These are the finite dimensional indecomposable rational linear representations of the additive group G_a of a field of characteristic zero, and their invariants are the kernel of the Weitzenbock derivation

Zero-separating invariants for linear algebraic groups

Let G be linear algebraic group over an algebraically closed field k acting rationally on a G-module V , and N(G,V) its nullcone. Let δ(G, V ) and σ(G, V ) denote the minimal number d, such that for any v ∈ V^G \ N(G,V) and v ∈ V \ N(G,V) respectively, there exists a homogeneous invariant f of positive degree at most d such that f (v) = 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V . For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL 2 (k) which contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G 0 is unipotent. Our results also lead to a more elementary proof that β_sep(G) is finite if and only if G is finite

On separating a fixed point from zero by invariants

Assume a fixed point v in V^G can be separated from zero by a homogeneous invariant f ∈ k[V]^G of degree p^r d where p > 0 is the characteristic of the ground field k and p, d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r , which we obtain explicitly from f . It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power

On the top degree of coinvariants

Cataloged from PDF version of article.For a finite group G acting faithfully on a finite-dimensional F-vector space V, we show that in the modular case, the top degree of the vector coinvariants grows unboundedly: lim(m ->infinity) topdeg F[V-m](G) = infinity. In contrast, in the nonmodular case we identify a situation where the top degree of the vector coinvariants remains constant. Furthermore, we present a more elementary proof of Steinberg's theorem which says that the group order is a lower bound for the dimension of the coinvariants which is sharp if and only if the invariant ring is polynomial

Separating Invariants for the Klein Four Group and Cyclic Groups

Cataloged from PDF version of article.We consider indecomposable representations of the Klein four group over a field of characteristic 2 and of a cyclic group of order pm with p, m coprime over a field of characteristic p. For each representation, we explicitly describe a separating set in the corresponding ring of invariants. Our construction is recursive and the separating sets we obtain consist of almost entirely orbit sums and products. © 2013 World Scientific Publishing Compan

From Jane Austen to Meghan Markle: The Persistence of British Imperialism in White Popular Feminism

This dissertation traces the persistent threads and values of white womanhood from the nineteenth-century British Empire to modern American popular culture. The figure of the white woman was significant to upholding colonialism and empire in the literary mass media and culture of the nineteenth century, and I argue that this figure continues to be used in popular media and online content today to surreptitiously uphold white supremacy and obscure race and gender inequalities. This dissertation will explore the overlaps between nostalgia, historical revisionism, white womanhood, white supremacy, and white feminism in modern American popular culture. The connections between, and the popularity of this broader media is not accidental but part of a longer history of white supremacy using culture and women to surreptitiously reinforce hierarchies and establish white-centered norms. This dissertation builds on work on white popular feminism, white womanhood, and cultural ideologies from scholars like Sarah Banet-Weiser, Koa Beck, Jessie Daniels, and Rafia Zakaria, while reflecting on how Black and intersectional feminisms, articulated by Kimberlé Crenshaw, Angela Davis, Patricia Hill Collins, and Audre Lorde among others, offer more revolutionary and effective forms of feminism and empowerment. From the prolific and consistent remediation of Jane Austen and her centurial contemporaries to the obsession and controversies surrounding Meghan Markle’s inclusion in, and subsequent exclusion from, the British Royal Family, this dissertation takes seriously the often overlooked and dismissed media and popular culture made for and by women to trace the histories of empire and their entanglement with a white popular feminism and white supremacy. Chapter one analyzes the popularity and reception of period media from 2020, Bridgerton, Emma., and Enola Holmes, to explore how period media, even those that attempt to be diverse and more progressive, still cultivate a white nostalgia for a past that aligns with a popular, white feminism that is non-threatening towards capitalism and white supremacy. Chapter two uses two popular remediations of Jane Austen’s novels, Clueless and Bridget Jones’s Diary, to trace the combination of Jane Austen and period media with postfeminism. This fusion embedded nineteenth-century values of white womanhood into popular feminist media that continues to have influence today. Chapter three will use the media surrounding Meghan Markle’s in/exclusion from the British Royal Family as demonstrating the promise and influence of a white popular feminism beyond fictional narratives, but also its limitations and failures when it goes against white supremacist patriarchal systems. The conclusion will then briefly extend the argument made throughout the chapters into social media spaces to connect how historical fantasy, urban homesteading, and constant cycles of trendy femininity reflect the white popular feminism and romanticization of imperial womanhood online. This dissertation takes seriously the narratives of idealized white womanhood that extend through recent centuries, and while media like Bridgerton and Enola Holmes may make it seem like a distant past, these imperial values of white womanhood are very much still present and influential within white popular feminism that guides larger discussions of inequality, justice, and white supremacy in our present moment

The Cohen-Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded sep- arating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separat- ing algebra implies the group is generated by bireflections. Ad- ditionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay