All Questions
Tagged with kt.k-theory-and-homology rt.representation-theory
51 questions
3
votes
0
answers
122
views
Canonical basis in equivariant K-theory of the Springer resolution
In Definition 15.0.2 of the notes from a course by Bezrukavnikov there is a characterization of canonical basis in K-theory of a Springer fiber which is due to Lusztig. This characterization is in ...
- 3,054
5
votes
1
answer
144
views
Equivariant KR-theory of representation sphere
I would like to say my question first.
Let $G$ be a compact Lie group acting on a good space $X$ in a good way. Let $V$ be a $G$-representation whose real dimension may be less than 8, and let $S^V$ ...
CommunityBot
- 1
10
votes
0
answers
225
views
Third homology of simply connected Chevalley–Demazure group schemes
I’ve been studying the group of $\mathbb{Z}$-points of the simply connected Chevalley–Demazure group scheme of type $E_7$, denoted $G_{\text{sc}}(E_7,\mathbb{Z})$; see Vavilov and Plotkin - Chevalley ...
- 545
1
vote
0
answers
124
views
Computing the induced homomorphisms of derived functors using acyclic resolutions
Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\...
- 38.6k
3
votes
0
answers
85
views
Explicit computation of the transfer in the representation ring for unitary groups
For a compact Lie group $G$ we let $R(G)$ be the ring of finite dimensional complex $G$-representations studied by Segal in http://www.numdam.org/item/PMIHES_1968__34__113_0.pdf.
This comes with extra ...
- 73
20
votes
2
answers
3k
views
Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory
First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex $((...
- 6,162
2
votes
0
answers
134
views
Algebra of finite width matrices
$\DeclareMathOperator\FWM{FWM}\DeclareMathOperator\End{End}$For any ring $R$ there's an algebra of finite width matrices with entries in $R$. By finite width matrices I mean the ones that have only ...
- 63.9k
4
votes
1
answer
204
views
"Ring of traces" over a ring R
$\DeclareMathOperator\RoT{RoT}$I'm interested in the following ring. Fix a (Noetherian?) base ring $R$, and consider the category of finitely generated projective $R$-modules equipped with ...
- 12.9k
7
votes
0
answers
270
views
The Todd class and Weyl's character formula
Let $\mathfrak{g}$ be a finite-dimensional complex semi-simple Lie algebra. Fix a Cartan sub algebra $\mathfrak{h} \subset \mathfrak{g}$ and let $R \subset \mathfrak{h}^{\ast}$ denote the root system. ...
- 1,379
8
votes
2
answers
208
views
Generalisation of the equivalence between $C^*(H)$ and $C_0(G/H) \rtimes G$; induction of group actions on C*-algebras
There is a well known Morita equivalence between the group C*-algebra $C^*(H)$ and $C_0(G/H) \rtimes G$, where $H$ is a subgroup of $G$. The corresponding equivalence of representations is an ...
- 12.9k
1
vote
0
answers
151
views
"Interesting" examples of exact abelian subcategories of R-Mod
A somewhat vague question: for which rings there exist "interesting" exact abelian subcategories of $R-\operatorname{Mod}$ that are closed with respect to products? Actually, I would like ...
- 16.9k
2
votes
1
answer
534
views
Kasparov's Dirac element and the index map
In Kasparov's 1988 paper Equivariant KK-theory and the Novikov conjecture section 4 he defined the Dirac element for a (non-spin) $G$- Riemanian manifold $X$ as an element in the $K$-homology $K^0_G(...
- 1,446
6
votes
0
answers
201
views
Hall-Littlewood polynomials of non-dominant weights
$\DeclareMathOperator\SL{SL}$Let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a sequence of positive integers and let
$$
R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
- 528
2
votes
0
answers
127
views
When semi-simple subcategories "extend" to hearts of t-structures?
Let $A$ be a semi-simple abelian subcategory of a triangulated category $C$ that "generates" $A$ (that is, $C$ equals its own smallest triangulated subcategory that is closed under direct ...
- 16.9k
35
votes
5
answers
6k
views
Understanding a quip from Gian-Carlo Rota
In the chapter "A Mathematician's Gossip" of his renowned Indiscrete Thoughts, Rota launches into a diatribe concerning the "replete injustice" of misplaced credit and "forgetful hero-worshiping" of ...
- 10.5k
2
votes
0
answers
147
views
About the algebraic structure of the $G$-equivariant $KK$-theory
Let $ G $ be a second countable locally compact group.
Let $ A $ and $ B $ be two $G$-$C^*$-algebras.
Let $ KK^G (A, B) $ be the $G$-equivariant $KK$-theory of the pair $ (A, B) $.
Could you tell me ...
- 35.5k
6
votes
1
answer
303
views
Irreducible representations of the symmetric group on homology of simplicial complex
I am following Wall's paper A note on symmetry of singularities and I have some questions regarding representation theory and the homology of some objects:
Consider an action of $\Sigma_k$ on a finite ...
CommunityBot
- 1
4
votes
0
answers
300
views
Is there algebraic $K$-theory of a group independent of the base ring?
Given a ring R and a group $G$, I can consider the group ring $R[G]$ and then take the algebraic $K$-theory $K(R[G])$. This the $K$-theory of the category $\operatorname{Rep}_R(G)$. As a variant, one ...
- 15.4k
3
votes
0
answers
206
views
About the representation ring of a compact group
A question stuck in my mind when I was reading the paper "The representation ring of a compact Lie group" by Segal. He says on page one that I confine myself to the case of a compact Lie ...
- 63.9k
6
votes
1
answer
752
views
Locally trivializing a G vector bundle?
In §1.6 of Atiyah's K-theory, he defines the notion of a $G$-(vector)-bundle, which is a sort of "equivariant vector bundle" with respect to a finite group action. More specifically, let $G$ ...
- 11.1k
8
votes
1
answer
234
views
What is the inverse in K-theory represented by Clifford module extensions?
I am working on a model for topological KO-theory which is represented by explicit spaces of orthogonal Clifford module extensions. That is, assuming $M$ compact, $KO^{-n+1}(M) := [M,X_n]$ where the ...
- 151
4
votes
0
answers
152
views
How much vanishing of odd K-groups implies the vanishing of odd equivariant K-groups?
The main quetion is
For a compact Lie group $G$, and a $G$-space $X$ with $K^1(X)=0$.
How much can we say about the vanishing of $K_G^1(X)$? Moreover, how much $K^0_G(X)=K^0(X)\times R(G)$?
Here $...
- 719
3
votes
1
answer
446
views
A set of objects classically generates the full subcategory of compact objects iff it generates the whole category
Sorry in advance if my question doesn't have the level of this community.
I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated ...
- 3,784
4
votes
1
answer
208
views
Bott periodicity homeomorphisms for spaces of Clifford extensions
I am trying to prove the following statement of real Bott periodicity, on the level of actual spaces of Clifford module extensions (i.e., not equivalence classes of modules).
Let $W = \mathbb{R}^{\...
- 63.9k
7
votes
1
answer
345
views
When and why are Adams operations "non-negative"?
We can think of the unary operations in a lambda-ring as integer linear combinations of Young diagrams; for example the operation $\lambda^n$ corresponds to the Young diagram with $n$ rows and one ...
CommunityBot
- 1
11
votes
0
answers
930
views
Higher traces in Hochschild cohomology
Let $A$ be an associative algebra over a field $k$. Let $\rho:A \rightarrow \mathrm{End}(M)$ a left module, finite dimensional over $k$. Then the map $a \mapsto \mathrm{tr}_M \rho(a)$ is a well ...
- 63.9k
4
votes
0
answers
212
views
When does the canonical $t$-structure restrict to perfect complexes?
I am interested in non-Noetherian(!) rings such that the canonical $t$-structure on $D(R)$ (the derived category of left $R$-modules) restricts to perfect complexes i.e. to the subcategory of ...
- 16.9k
4
votes
1
answer
470
views
On definitions and explicit examples of pure-injective modules
I am interested in the following assumption on left $R$-modules: for a module $I$ and all injective homomorphisms $A\to B$ of finitely generated (or possibly finitely presented) modules I want the ...
- 16.9k
5
votes
0
answers
87
views
Existence of anti-symmetric hochschild homology representatives
Let $A$ be an associative algebra over a field $k$. Let $A_{L}$ be the Lie algebra of $A$ with commutator bracket. Then if $M$ is a bimodule for $A$ there is an associated representation of $A_{L}$ ...
CommunityBot
- 1
6
votes
2
answers
487
views
Induction theorems for finite-dimensional complex representations of infinite groups
Let $G$ be a group, usually infinite. I am interested in finite-dimensional complex unitary representations of $G$, i.e. group homomorphisms $G \rightarrow U_n(\mathbb{C})$. The category of these ...
- 12.9k
37
votes
0
answers
922
views
Chern character of a Representation
Let $G$ be a finite group. Under the identification of the representation ring $R_{\mathbb{C}}(G)$ with the equivariant K-theory $KU^0_G(\ast)$ of the point, followed by Atiyah-Segal completion-...
- 19.8k
2
votes
1
answer
451
views
$SO(6) \to SU(2) \times SU(2) \times U(1)$ branching rules
What do these branching rules mean?
\begin{eqnarray*} SO(6)_E &\to& SU(2)_\ell \times SU(2)_r \times U(1)_\Sigma
\end{eqnarray*}
I am taking these examples from a paper of Gukov (on p.51) ...
CommunityBot
- 1
4
votes
0
answers
152
views
Character Theory for quaternionic representations of finite groups
Is there a nice character Theory for quaternionic representations of finite groups?
By this I mean a description of the number of Quaternionic representations of a finite ...
- 2,630
5
votes
1
answer
392
views
Equivariant $K$-theory, singular vectors, and flag manifolds
For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ ...
- 765
9
votes
2
answers
583
views
Simplest explicit counterexample for $Vect(BG) \ne Rep(G)$ as monoids
Let $G$ be a topological group, $Vect(BG)$ the monoid of complex vector bundles over its classifying space (not the stack!) and $Rep(G)$ its monoid of complex representations.
Generally $Vect(BG) \ne ...
- 56.9k
15
votes
1
answer
758
views
Swan K-theory of Z/4
Given a finite group $G$ and a commutative ring $R$, define the Swan $K$-theory $K_0(G, R)$ to be the Grothendieck group of the category finitely generated projective $R$-modules with $G$-action (with ...
- 35.2k
4
votes
0
answers
175
views
Seeking an unpublished manuscript by Tetsuro Okuyama
Several papers in representation theory attribute the notion of relatively projective modules to Tetsuro Okuyama's manuscript "A generalization of projective covers of modules over finite group ...
- 30.3k
3
votes
0
answers
599
views
Representations and K-theory of a finite group
This question is motivated by the calculation of the higher algebraic $K$-groups of a finite field.
Let $G$ be a finite group, the case I am most interested in is $G = \text{Gl}_n(\mathbb F_q)$, but ...
- 11.9k
9
votes
1
answer
579
views
Heller operator without left adjoint?
Suppose given a noetherian ring $R$. On the stable category $R\text{-}\underline{\text{mod}} := R\text{-mod}/R\text{-proj}$, we have the Heller operator
$$
\Omega : R\text{-}\underline{\text{mod}} \...
CommunityBot
- 1
4
votes
1
answer
592
views
What sort of ring-theoretic properties does the representation ring of a compact Lie group possess?
Recall the definition of the representation ring $R(G)$ of a compact Lie group $G$. I'd like a reference that gives me basic ring-theoretic properties that $R(G)$ always has, or enough info that I can ...
- 118k
7
votes
1
answer
615
views
Representation ring and induced representation
Let $i:H \to G$ be a homomorphism of compact Lie groups. The induced representation $\iota_*V := \mathrm{Map}^H(G,V)$ of an $H$-representation $V$ does not give an element of the representation ring $...
- 811
0
votes
1
answer
340
views
Length of a module
Let R be a commutative ring, M an R-module of finite length and let N be an Injective R-module with zero socle. Then why $ \text{Hom}_R(M, N) $ is zero?
CommunityBot
- 1
0
votes
0
answers
165
views
Cotorsion theory and its relative homology
Let (F(R), Cot(R)) be a cotorsion theory, Such that F(R) is the class of flat R-modules and Cot(R) the cotorsion modules. Why this is true that, For $ N\in Cot(R) $,
$ \text{Ext}_{F(R)}^i(M, N)\cong \...
- 11
0
votes
1
answer
330
views
About regular local rings and Socles
Let R be a regular local ring with $ \text{dim} R = d $. If $ 0\rightarrow R\rightarrow I_0\rightarrow ...\rightarrow I_d\rightarrow 0 $. Then why for $ 0\leq i\leq d-1 $, the socle of $ I_i $ is ...
- 672
4
votes
0
answers
323
views
The proof of the splitting principle in equivariant K-theory via flag manifolds
In Atiyah's famous paper "Bott periodicity and the index of elliptic operators" section 4, he proved the splitting principle for unitary groups (Propostion 4.9 in that paper), namely:
Let $j: T\...
- 8,559
6
votes
1
answer
288
views
Projective modules over non-rational group rings
Let $G$ be a finite group. We know that the $K$-group $K_0(QG)$ of the rational group ring $QG$ is a free abelian group generated by the irreducible representations of $G$ over $Q$. Now let $R$ be a ...
- 11.1k
2
votes
0
answers
263
views
k-theory of $\mathbb{Z}$
I have a doubt.
Borel computed the rank of the higher algebraic k-theory of $\mathbb{Z}$:
$rank(K_n)(\mathbb{Z})= 1$ if $n\equiv1 mod4$, otherwise this rank is equal to 0.
On the other hand Bjorn ...
- 235
1
vote
1
answer
318
views
Can injective resolutions be 'enlarged' (or shrunk) to admit only injective maps from extensions?
Let $M$ and $N$ be $R$-modules for some ring $R$. There is a standard result involving the computation of $\text{Ext}^n(M,N)$, using projective resolutions, which says that you can always choose a ...
- 666
3
votes
2
answers
312
views
Extensions which define the same element of $\text{Ext}^n(M,N)$ are in fact equivalent
It is well known (and wouldn't be so-named unless it were) that:
If $\xi$, $\eta$ are $n$-fold extensions of $N$ by $M$ (modules over a ring $R$) which yield the same element of $\text{Ext}^n(M,N)$, ...
- 2,721
6
votes
1
answer
465
views
Splitting principle in equivariant cohomology
The following is a weaker version of what is called splitting principle in
Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6:
Let $G$ be a compact ...
- 27k