# Tagged Questions

**12**

votes

**0**answers

214 views

### Adams Operations on $K$-theory and $R(G)$

One of the slickest things to happen to topology was the proof of the Hopf invariant one using Adams operations in $K$-theory. The general idea is that the ring $K(X)$ admits operations $\psi^k$ that ...

**2**

votes

**1**answer

455 views

### 1D TQFT in Freed-Hopkins-Lurie-Teleman

In the first section of Freed-Hopkins-Lurie-Teleman they construct a one-dimensional Topological Quantum Field Theory.
$F(\circ_+)$ is a vector space and $F(\circ_-)$ is the dual.
$F(\circ-\circ)$ ...

**4**

votes

**2**answers

389 views

### What does Freed-Hopkins-Teleman say about finite groups?

The third in the "Loop groups and twisted K-theory" series by FHT treats compact Lie groups without any connectedness assumptions. I am trying to unwind what Theorem 2 of that paper (available here, ...

**2**

votes

**0**answers

108 views

### Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...

**3**

votes

**1**answer

146 views

### Condition on a Hopf operad for tensor product in the base categoy to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...

**16**

votes

**2**answers

538 views

### What's the relationship between these two isomorphisms involving G and T?

Let $G$ be a compact connected Lie group with maximal torus $T$ and Weyl group $W$. Recall the following two isomorphisms.
Isomorphism 1: $R(G) \cong R(T)^W$, where $R(-)$ denotes the representation ...

**3**

votes

**0**answers

97 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 $V$, ...

**7**

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**0**answers

228 views

### Characteristic Classes in Geometric Representation Theory

Geometric respectively topological methods are widely applied in representation theory. As far as I know mainly cohomological methods are used.
I wonder if there are concrete applications of the ...

**3**

votes

**1**answer

171 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**

votes

**1**answer

150 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 ...

**9**

votes

**2**answers

395 views

### Cohomology ring of a flag variety and representation theory

I'm interested in the cohomology ring $H^*(G/B)$ of a flag variety $G/B$, where $G$ is a complex semi-simple Lie group and $B$ the Borel subgroup. Borel (1953) showed that this ring is isomorphic to ...

**8**

votes

**2**answers

250 views

### Are torus knot groups linear?

The fundamental group $T(p,q)$ of the complement of a $(p,q)$-torus knot (in $S^3$) admits the presentation $\langle a, b \mid a^p=b^q \rangle $. Is $T(p,q)$ linear, i.e., is there a faithful ...

**12**

votes

**3**answers

730 views

### Why do we need a $G$-universe?

Let $G$ be a compact Lie group. Before defining $G$-prespectra, we have to define a $G$-universe $\mathcal U$.
Question: Why do we need a $G$-universe?
A $G$-universe is defined to be a countably ...

**3**

votes

**0**answers

120 views

### How to understand specialization in Borel-Moore homology

In Chriss-Ginzburg, they defined specialization in Borel-Moore homology on page 103. I think the definition is too abstract and it's hard for computation. Is there any way to understand it better?
...

**4**

votes

**1**answer

199 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 ...

**7**

votes

**1**answer

218 views

### What's a good example/reference for cohomology classes on Springer fibers that aren't restricted from the flag variety

As usual, by Springer fiber, I mean the fixed points $X^u$ of a unipotent element $u$ of the group $G$ on the flag variety $X=G/B$. It's a lovely theorem that when $G=SL_n$, the induced map on ...

**0**

votes

**1**answer

173 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 ...

**2**

votes

**1**answer

263 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**

votes

**0**answers

98 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 ...

**2**

votes

**2**answers

277 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 ...

**2**

votes

**2**answers

373 views

### Equivariant Cohomology of a Complex Projective Variety

Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...

**4**

votes

**1**answer

483 views

### Dijkgraaf-Witten TQFT vs. Representation Theory?

From what I had read, group characters can be "glued" together in a topological fashion and there is something to this effect in the paper by Dijkgraaf and Witten. TQFT seems to be a topological ...

**3**

votes

**2**answers

557 views

### Quotients in Sums of Rings

Suppose we are given a commutative ring R with unit-element. Now we have a composition of R as the direct product of two rings $R\cong R_1\times R_2$. It is now straight forward, that any ideal ...

**8**

votes

**3**answers

715 views

### HIgher Homotopy Groups and Representation Theory

Let $G$ be a compact Lie group, and $g$ its associated Lie algebra.
In what ways do the higher homotopy groups $\pi_{n}(G)$ with $n>1$ appear in the representation theory of $G$?
As an example, ...

**2**

votes

**2**answers

226 views

### Stabilizers for Nilpotent Adjoint Orbits of Semisimple Groups

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (ie. that $ad_X:\frak{g}\rightarrow\frak{g}$ is ...

**3**

votes

**1**answer

206 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**

votes

**1**answer

149 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 ...

**5**

votes

**1**answer

261 views

### Representations of SO(3) and vector bundles on BSO(3)

Let $V$ be the vector bundle over $BSO(3)$ associated to the adjoint representation of $SO(3).$ Then $V$ does not have a nonzero section. One way to see this is that the Steifel-Whitney class ...

**8**

votes

**0**answers

257 views

### Chern Classes of Exterior Products of a vector bundle.

This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of ...

**10**

votes

**1**answer

287 views

### When is the derived category of representations of a finite poset equivalent to its opposite?

If I have a finite partially ordered set $K$, I can look at its derived category of finite dimensional representations $D(K)$. Note that $D(K^{op}) \simeq D(K)^{op}$ by linear duality.
But when do ...

**6**

votes

**0**answers

523 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 ...

**11**

votes

**3**answers

519 views

### Effect on homology of decorating vertices of a simplicial complex

In my research, the following construction came up.
Let $X$ be an $n$-dimensional simplicial complex. For an integer $m \geq 1$, let $X[m]$ denote the following simplicial complex. The vertices of ...

**2**

votes

**0**answers

204 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 ...

**6**

votes

**2**answers

234 views

### equivariant cohomology of the complement to the arrangment $\cup_{i\neq j}overrightarrow{x_i} = overrightarrow{x_j}$?

Let $V=\mathbb{R}^d$ be a $d$-dimensional (Euclidian) vector space over real numbers.
Let $G=SO(V)$ be a compact Lie group of linear orthogonal transformations of $V$.
Let $Conf_n(V)$ be the space of ...

**2**

votes

**1**answer

259 views

### Cell decomposition for a variety not necessarily complete?

Let $X$ be an algebraic variety with a $\mathbb C^*$ action such that the fixpoints set is finite. By theorem 4.3 in the paper of Bialynicki-Birula "Some theorems on actions of algebraic groups", ...

**4**

votes

**0**answers

141 views

### Whitehead products and a realization problem for graded Lie algebras

Many $\mathbb{Z}$-graded Lie algebras $\mathfrak{g}$ over $\mathbb{C}$ we would like to study are non-degenerate in the sense that
$\dim_{\mathbb{C}} \mathfrak{g}_n < \infty \ \forall n \in ...

**3**

votes

**1**answer

238 views

### Kernel of the representation of the mapping class group to $Aut(F_n)$

Let $S_{g,1}$ be a orientable compact surface of genus $g$ with one boundary component and $\Gamma_{g,1}$ the mapping class group.
By $F_n$ I denote the free group on $n$ generators.
One obtains a ...

**6**

votes

**5**answers

1k views

### Weight lattice and the first fundamental group

Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively. Then, $\frak{t}$ can be considered as a Cartan subalgebra of ...

**9**

votes

**4**answers

907 views

### Coincidences amongst classifying spaces and Eilenberg Mac-Lane spaces

Given that $$\mathbb{R}P^{\infty} = B O(1) = K(\widehat{O(1)}, 1)$$ $$\mathbb{C} P^{\infty} = B U(1) = K( \widehat{U(1)}, 2)$$ is there any way to make sense of $$\mathbb{H}P^{\infty} = B Sp(1)$$ in a ...

**0**

votes

**1**answer

267 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 ...

**8**

votes

**2**answers

377 views

### Cohomology of representation varieties

Perhaps this question is too general then I am sorry about this.
My question is the following.
Let $\pi$ be the fundamental group of a compact surface of genus $g$ (with if necessary $n$ punctures) ...

**21**

votes

**8**answers

1k views

### Avatars of the ring of symmetric polynomials

I'm collecting different apparently unrelated ways in which the ring (or rather Hopf algebra with $\langle,\rangle$) of symmetric functions $Z[e_1,e_2,\ldots]$ turns up (for a Lie groups course I will ...

**9**

votes

**3**answers

913 views

### Why is the dual of a torus the same as its fundamental group?

The set of continuous homomorphisms from a torus ${\mathbb T}^n = ({\mathbb R}/{\mathbb Z})^n \to {\mathbb R}/{\mathbb Z}$ can be identified with ${\mathbb Z}^n$ if we assign to each $k = (k_1, \ldots ...

**16**

votes

**2**answers

1k views

### Virasoro action on the elliptic cohomology

I'm trying to understand better the mathematical notion of elliptic cohomology. Note that I only know the physics definition of the elliptic genus given in Witten's paper.
Let $X$ be a Calabi-Yau ...

**5**

votes

**2**answers

425 views

### explicit linear representations of fundamental groups of surfaces

I am looking for an explicit representation of the fundamental group of a closed orientable surface of genus >1. I guess they should be abundant in degree 2. Did anyone see the explicit matrix ...

**7**

votes

**3**answers

329 views

### Group Extensions and Line Bundles on $BG$

I am sure the answer to this question is well-known, but
It is well known that the group cohomology $H^2(G,\mathbb Z)$ classifies group extensions $0\to \mathbb Z\to E\to G\to 1$ and that for a ...

**14**

votes

**4**answers

1k views

### Poincare dual in equivariant (co)homology?

Let $G$ be a compact Lie group, $X$ be a (compact, oriented) smooth manifold, with $G$ acts on $X$ smoothly. Then we can talk about the $G$-equivariant homology and cohomology.
My question: In what ...

**3**

votes

**1**answer

530 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} = ...

**8**

votes

**2**answers

403 views

### Applications of classifying thick subcategories

So, relatively recently, Balmer introduced this notion of a spectrum for a tensor triangulated category and used it to prove a generalization of a classification theorem done in several areas of ...

**12**

votes

**3**answers

3k views

### Cohomology of Flag Varieties

For $K$ a compact Lie-group with maximal torus $T$, I'd like to know the cohomology $\text{H}^{\ast}(K/T)$ of the flag variety $K/T$.
If I'm not mistaken, this should be isomorphic to the algebra of ...