All Questions
Tagged with rt.representation-theory at.algebraic-topology
142 questions
4
votes
2
answers
1k
views
Why is the equivariant Euler class a character ?
Let us first precise the question : let $T$ be a torus, $\alpha : T \to \mathbb{C}$ be an irreducible character. I am interested in the $T$-equivariant Euler class of the ($T$-equivariant) bundle $\...
- 502
4
votes
1
answer
163
views
Representations of Finite Subgroups on Homology
Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting ...
- 4,920
4
votes
1
answer
184
views
FI-homology of a spectral sequence of rational FI-modules
Let $(E^r_{p,q})$ be a spectral sequence of rational $\mathsf{FI}$-modules. Call $H^{\mathsf{FI}}$ the $\mathsf{FI}$-homology (see here) and $t_k X= \deg H^{\text{FI}}_k X$ the $k$-th generation ...
- 275
4
votes
1
answer
236
views
Equivariant complex $K$-theory of a real representation sphere
Consider the one-point compactification of a $U(n)$-representation $V$, denoted by $S^V$. I want to caclulate $\tilde{K}_\ast^{U(n)}(S^V)$. When $V$ is a complex $U(n)$-representation, we can use the ...
user501794
4
votes
1
answer
370
views
Cohomology of Projective Classical Lie Groups
Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes ...
- 4,920
4
votes
1
answer
742
views
Restriction map for Lie algebra/Lie group cohomology associated to a complex semisimple Lie algebra and a semisimple Lie-subalgebra
Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra (or the corresponding Lie group). For definiteness, I'll take $\mathfrak{g}$ to be of type $A_n$, that is, $\mathfrak{g} = \...
- 2,160
4
votes
1
answer
193
views
Decomposition of $\Gamma$-modules into simple objects
Let $\Gamma$ be the category of finite pointed sets. The abelian category $\mathrm{Mod-}\Gamma$ is the category of functors $\Gamma^{\mathrm{op}} \to \mathrm{Vect}_k$, where $k$ is a field (see ...
- 5,482
4
votes
2
answers
473
views
Semidirect product of torus with cyclic group: representations/cohomology?
Let $p$ be prime and let $T^p$ be the $p$-torus and $\mathbb{Z}/p$ the cyclic group of order $p$ generated by $(12\ldots p)$. Consider the semidirect product $T^p\rtimes\mathbb{Z}/p$ with the natural ...
- 333
4
votes
1
answer
341
views
U(1) vs. BZ and representations of 2-groups
$U(1)$ seems to lead a dual life. On one hand it is the group we know and love, and on the other, it is the classifying space of the integers. Thinking about $n$-groups says that we should also think ...
- 631
4
votes
1
answer
299
views
Maurer-Cartan elements of the extension of an $L_{\infty}$-algebra
Let $g$ be a nilpotent $L_{\infty}$-algebra. For every commutative differential graded algebra $A$, one can form the extension $g\otimes A$ and endow it with a nilpotent $L_{\infty}$-algebra structure....
- 1,609
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}^{\...
- 151
4
votes
0
answers
128
views
Real Representation ring of $U(n)$ and the adjoint representation
I have two questions:
It is well known that the complex representation ring $R(U(n))=\mathbb{Z}[\lambda_1,\cdots,\lambda_n,\lambda_n^{-1}]$, where $\lambda_1$ is the natural representation of $U(n)$ ...
user340793
4
votes
0
answers
181
views
Specify the embedding of special unitary group in a Spin group via their representation map
How do we specify the embedding of a Lie group $G_1$ as a subgroup into a larger Lie group $G_2$, with $G_1 \subset G_2$ that agree with a constraint on the mapping between their representations?
By ...
- 10.5k
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
4
votes
0
answers
192
views
Extended double 2-cocycle conditions: Mathematical structure behind?
Note: For experts, to save your time, you can just read the highlighted texts and Eqs directly.
The ordinary group 2-cocycle condition:
Let us remind the usual so-called homogeneous group 2-cocycle $...
- 10.5k
4
votes
0
answers
182
views
'Noether normalization' for finite group schemes
Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$.
Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...
- 3,784
4
votes
0
answers
135
views
Exotic 2-adic lifts of mod $2$ Steinberg idempotent
Denote $B_n$ the Borel subgroup of $Gl_n(Z/2)$, i.e., the subgroup of
upper triangular matrices, $\Sigma _n$ the subgroup of permutation matrices.
The (conjugate) Steinberg idempotent is defined to be ...
- 3,051
3
votes
1
answer
238
views
Steenrod operations on classifying spaces
Steenrod operations can be defined for all finite characteristics $p$. The simplest one, when $p=2$, is the Steenrod square. I wonder if the computation for classifying spaces for classical Lie groups ...
- 803
3
votes
2
answers
704
views
Closure relations between Bruhat cells on the flag variety
Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...
- 1,719
3
votes
1
answer
251
views
About decomposition theorem BBD with respect to some stratification
I want to follow up a question from here (how to deduce version 1.a. from version 1).
I know a version of decomposition theorem BBD:
Version 1. Let $f:X\to Y$ be a (surjective) proper map of complex ...
- 133
3
votes
1
answer
136
views
symmetric group of regular polyhedrons
Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,
$\Delta^2$:
$\Delta^3$:
Let $c:=c(...
- 543
3
votes
1
answer
255
views
Projective representation of diffeomorphism group of $S^2$ [closed]
We know that the projective representation of a group $G$ is classified by $H_{grp}^2(G,R/Z) = H^3(BG,Z)$, where $H^*_{grp}$ is the group-cohomology class.
Then do we have a classification of the ...
- 4,826
3
votes
0
answers
249
views
Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)
In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as
$$D\; X^n = F(\tfrac{...
- 10.5k
3
votes
0
answers
547
views
Aut/Inn/Out Automorphism Groups of the unitary group $𝑈(𝑁)$
Given a group $G$, we denote the center Z$(G)$, we like to know the
automorphism group Aut($G$), the outer automorphism Out($G$) and the inner automorphism Inn($G$). They form short exact sequences:
$$...
- 10.5k
3
votes
0
answers
167
views
Obstruction to delooping
Let $G$ be a finite group. It can be think of as a $1$-category with one object and $|G|$ many morphisms. If $A$ happens to be abelian, then one can think of it to an $n$-category. Conversly, this ...
- 5,230
3
votes
0
answers
150
views
Integral Homology of GIT Quotients
Is there any example of GIT quotients of linear actions on a Euclidean space ${\mathbb C}^n$ that satisfy the following conditions?
The quotient is compact and smooth.
The homology of the quotient ...
- 1,207
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
3
votes
0
answers
117
views
Why "non-linear similarity" is the same as equivalence of representations for connected Lie groups?
Let $G$ be a compact Lie group and $V$ a finite-dimensional orthogonal $G$-representation. Write $S^V$ for the quotient $D(V)/S(V)$, where $D(V)$ and $S(V)$ are the unit disk and sphere in $V$, ...
2
votes
1
answer
235
views
Homotopy classes of homeomorphism vs. Homotopy classes of a biholomorphism
This is a more detailed question about my first question Representation theory and topology of Teichmüller space, I asked there how to understand:
$$T_{g}\hookrightarrow Hom(\pi_{1}({S}),PSL_{2}(\...
- 77
2
votes
1
answer
391
views
cohomology of orthogonal (or general linear) group over finite fields
Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let
$$
O(\mathbb{Z}_2^{\oplus k})=\{A\mid A \text{ is a } k\times k \text{ - matrix with entries } 0,1, det(A)=\pm 1\}
$$
What is $$
H^*(BO(\mathbb{Z}_2^{\...
- 1,990
2
votes
1
answer
306
views
connectedness of fibers of torus-equivariant moment maps
Given a possibly singular, connected, symplectic algebraic variety with a torus action, every fiber of the moment map admits a torus action. Is each fiber of this moment map connected? Any examples or ...
- 1,719
2
votes
1
answer
687
views
Derived Push-Forward of Morphism of Perverse Sheaves and Translation Functors
I hope this question is not too vague.
Let $G$ be a complex reductive group, $B$ a Borel subgroup of $G$, and $P$ a parabolic containing $B$.
Denote by $\pi:G/B\to G/P$ the canonical map. Consider ...
- 2,554
2
votes
0
answers
169
views
How a circle $S^1$ acts on the Cayley plane $OP^2$ with exactly three fixed points?
The complex projective $CP^2$, the quaternionic projective space $HP^2$, and the octonionic projective space $OP^2$ each admit a circle action with $3$ fixed
points.
The circle action on $HP^2$ can be ...
- 103
2
votes
0
answers
169
views
The dimension of the representation ring
Let $G$ be a compact Lie group. I am trying to characterize the algebraic properties of the representation ring $R(G)$ of $G$. In the case of the $n$-torus, the representation ring $R(T)$ is ...
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
614
views
cohomology of orthogonal group of integers
Let
$$
O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k).
$$
What is $$
H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})?
$$
If it cannot be computed out, can we get
$$
H^*(O(\mathbb{Z}^{\oplus ...
- 1,990
1
vote
1
answer
248
views
Topology of a Compact Space with Fixed-Point-Free Torus Action
Let $X$ be a compact connected smooth manifold and $T$ a compact torus acting smoothly on $X$ without fixed points. What, in general, can be said about the topology of $X$ (ex. rational (co-)homology)?...
- 4,920
1
vote
1
answer
438
views
Categories associated to digraphs
Let's take a directed graph, or a digraph, $G=(V,E)$ given by a finite set of vertices $V$ and a finite set of edges $E$. We can assume that pairs of vertices can have parallel edges between them and ...
1
vote
0
answers
92
views
How are representations of SU(2) identified with non-intersecting curves in a disk? (Rumer-Teller-Weyl)
Apparently (e.g. see bottom left of first column on p3 of this manuscript) the representations of $SU(2)$ can be identified with diagrams of non-intersecting curves (unoriented embedded $1$-manifolds) ...
- 607
1
vote
0
answers
120
views
Computing $\text{Tor}_*^{R_G} (\mathbb{Z}, \mathbb{Z}) $ for a compact Lie group $G$
Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...
- 257
0
votes
1
answer
386
views
Reference request for equivariant cohomology of G [duplicate]
Possible Duplicate:
What is the equivariant cohomology of a group acting on itself by conjugation?
Let $G$ be a compact Lie group. Where can one read about the equivariant cohomology $H_G^*(G)$, ...
- 559
-2
votes
1
answer
516
views
no classification of nilpotent lie groups
there is no classification of (simply connected) nilpotent lie groups, but I am tempted to try to generalize the construction of the Heisenberg group. For an upper triangular matrix:
$$ \left(
\...
- 22.8k