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Blocks of the Grothendieck ring of equivariant bundles on a finite group  [PDF]
Cédric Bonnafé
Mathematics , 2014,
Abstract: If $G$ is a finite group, the Grothendieck group ${\mathbf{K}}\_G(G)$ of the category of $G$-equivariant ${\mathbb{C}}$-vector bundles on $G$ (for the action of $G$ on itself by conjugation) is endowed with a structure of (commutative) ring. If $K$ is a sufficiently large extension of ${\mathbb{Q}}\_{\! p}$ and ${\mathcal{O}}$ denotes the integral closure of ${\mathcal{Z}}\_{\! p}$ in $K$, the $K$-algebra $K{\mathbf{K}}\_G(G)=K \otimes\_{\mathbb{Z}} {\mathbf{K}}\_G(G)$ is split semisimple. The aim of this paper is to describe the ${\mathcal{O}}$-blocks of the ${\mathcal{O}}$-algebra ${\mathcal{O}} {\mathbf{K}}\_G(G)$.
On a rationality question in the Grothendieck ring of varieties  [PDF]
Hélène Esnault,Eckart Viehweg
Mathematics , 2009,
Abstract: This is a small note meant to be published in a Conference Proceedings. We discuss elementary rationality questions in the Grothendieck ring of varieties for the quotient of a finite dimensional vector space over a characteristic 0 field by a finite group. Part of it reproduces the content of a letter dated September 27, 2008 addressed to Johannes Nicaise
Equivariant $K$-theory of flag varieties revisited and related results  [PDF]
V. Uma
Mathematics , 2011,
Abstract: In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation ring of the torus $T$. We further apply our results to describe the multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the wonderful compactification of the adjoint group of $G$, in terms of the structure constants of Schubert varieties in the Grothendieck ring of $G/B$.
Conics in the Grothendieck ring  [PDF]
János Kollár
Mathematics , 2003,
Abstract: This note describes the subring of the Grothendieck ring of k-varieties generated by smooth conics; finding many zero divisors. The proof uses only elementary projective geometry.
Algebraic independence in the Grothendieck ring of varieties  [PDF]
N. Naumann
Mathematics , 2004,
Abstract: We give sufficient cohomological criteria for the classes of given varieties over a field $k$ to be algebraically independent in the Grothendieck ring of varieties over $k$ and construct some examples.
On the Grothendieck ring of varieties  [PDF]
Amit Kuber
Mathematics , 2013, DOI: 10.1017/S0305004115000079
Abstract: Let $\operatorname{K}_0(\operatorname{Var}_k)$ denote the Grothendieck ring of $k$-varieties over an algebraically closed field $k$. Larsen and Lunts asked if two $k$-varieties having the same class in $\operatorname{K}_0 (\operatorname{Var}_k)$ are piecewise isomorphic. Gromov asked if a birational self-map of a $k$-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. If these two questions admit a positive answer, then we prove that its underlying abelian group is a free abelian group. Furthermore, if $\mathfrak B$ denotes the multiplicative monoid of birational equivalence classes of irreducible $k$-varieties then we also prove that the associated graded ring of the Grothendieck ring is the monoid ring $\mathbb Z[\mathfrak B]$.
A local ring such that the map between Grothendieck groups with rational coefficient induced by completion is not injective  [PDF]
Kazuhiko Kurano,Vasudevan Srinivas
Mathematics , 2007,
Abstract: In this paper, we construct a local ring $A$ such that the kernel of the map $G_0(A)\subq \to G_0(\hat{A})\subq$ is not zero, where $\hat{A}$ is the comletion of $A$ with respect to the maximal ideal, and $G_0()\subq$ is the Grothendieck group of finitely generated modules with rational coefficient. In our example, $A$ is a two-dimensional local ring which is essentially of finite type over ${\Bbb C}$, but it is not normal.
Discriminants in the Grothendieck Ring  [PDF]
Ravi Vakil,Melanie Matchett Wood
Mathematics , 2012, DOI: 10.1215/00127094-2877184
Abstract: We consider the "limiting behavior" of *discriminants*, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we conjecture that the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and propose a number of new conjectures, both arithmetic and topological.
The equivariant cohomology rings of Peterson varieties  [PDF]
Yukiko Fukukawa,Megumi Harada,Mikiya Masuda
Mathematics , 2013, DOI: 10.2969/jmsj/06731147
Abstract: The main result of this note is an efficient presentation of the $S^1$-equivariant cohomology ring of Peterson varieties (in type $A$) as a quotient of a polynomial ring by an ideal $\mathcal{J}$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety. Our result simplifies previous presentations given by Harada-Tymoczko and Bayegan-Harada. In particular, our result gives an affirmative answer to a conjecture of Bayegan and Harada that the defining ideal $\mathcal{J}$ is generated by quadratics.
On generating series of classes of equivariant Hilbert schemes of fat points  [PDF]
S. M. Gusein-Zade,I. Luengo,A. Melle Hernandez
Mathematics , 2009,
Abstract: In previous papers the authors gave formulae for generating series of classes (in the Grothendieck ring of complex quasi-projective varieties) of Hilbert schemes of zero-dimensional subschemes on smooth varieties and on orbifolds in terms of certain local data and the, so called, power structure over the corresponding ring. Here we give an analogue of these formulae for equivariant (with respect to an action of a finite group on a smooth variety) Hilbert schemes of zero-dimensional subschemes and compute some local generating series for an action of the cyclic group on a smooth surface.
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