All Questions
931 questions with no upvoted or accepted answers
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What total orders have people studied on Coxeter Groups?
I'm aware of the ShortLex total order that gives rise to the usual normal form. But are there any others that have naturally arose and people have studied?
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268
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The double exponential map and the Baker–Campbell–Hausdorff–Dynkin series
$\DeclareMathOperator\Exp{Exp}$A. Gavrilov has several nice works studying the double exponential map and its properties ([1] and references therein).
Given a complete Riemannian manifold $(M,g)$ and ...
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3
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197
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How to write down an explicit equation of given degree yielding a smooth hypersurface in a projective space?
Let F be a field of positive characteristic $p$ and let $d,n$ be two positive integers.
Can we explicitly write down an equation defining a smooth hypersurface $X_d⊂\mathbb P^n_F$ of degree d ?
This ...
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3
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81
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The Loday-Quillen-Tsygan theorem for topological (Fréchet) algebras
In "Additive K-theory" by Tsygan and Feigin, Section 0.4, a statement is given which seems to generalize (cohomological version of) the well-known Loday-Quillen-Tsygan theorem
$$H_{\text{CE}}...
user126256
3
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79
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Multiplicity relation between highest weight modules, Demazure modules, and crystals
Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra, and let $\lambda$ be an associated dominant integral weight. Then two different objects we can relate to this data is $V(\lambda)$, the ...
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3
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85
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Couniversality of Lie integration in different categories of manifolds/smooth spaces
A fairly reasonable interpretation of Lie II and Lie III seems to be that the category of Lie algebras is a coreflective subcategory of the category of Lie groups, so that the Lie group integrating a ...
- 1,253
3
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218
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Is there a proper smooth variety in characteristic $p$ whose Hodge-to-de Rham spectral sequence does not degenerate at $E_1$?
By Deligne-Illusie, such a variety has no lifting to $W_2(k)$. In their paper they state that they do not know if such an example exists. Has this question been answered since then?
- 4,164
3
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137
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Generic linear subspaces of symmetric matrices
Let $\mathcal{S}_{n}(\mathbb{R})$ be the real vector space of symmetric $n\times n$ traceless matrices with real entries and let $L\subset \mathcal{S}_{n}(\mathbb{R})$ be a linear subspace. Noticing ...
- 3,020
3
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98
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Semisimple subgroup of Euclidean group
Let $G$ be a closed and connected semisimple subgroup of the Euclidean group $E(n)$ (the group of isometries of $\mathbb R^n$).
Can we prove that $G$ is conjugate to a subgroup of $O(n)$?
- 2,535
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123
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Jacobson-style Galois theory on perfect closure
Promoted from stack.exchange since I received no response:
Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots) $K/k$, where he ...
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3
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245
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On the Gelfand-Kirillov Conjecture
The base field $k$ is of zero characteristic.
Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is ...
- 3,318
3
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55
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Natural appeareances of (commutative) algebras in $\mathfrak g$-modules
$\newcommand{\g}{\mathfrak g}$
Let $\g$ be a Lie algebra, and observe that since $U(\g)$ is a cocommutative Hopf algebra, it makes sense to look for (naturally arising and perhaps commutative?) ...
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252
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Reductive Lie groups and existence of maximal compact subgroup
I am reading Knapp's book "Lie groups beyond an introduction" (2nd edition). I am struggling to understand the following point. Recall that $G$ is a reductive Lie group. If the Lie algebra $\...
- 1,879
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68
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Family of Lie algebras parametrized by a discrete valuation ring
I have a family of Lie algebras parametrized by a discrete valuation ring, whose generic fiber is reductive and whose special fiber is nilpotent. I'd like to learn about the relationship between the ...
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3
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95
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Decomposing a compact connected Lie group
I want to prove the following. Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ and center $Z_G.$ It is not hard to prove that $\mathfrak g$ is reductive. Therefore, we can ...
- 1,879
3
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126
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Nakajima reflection functors and the flavour/framing group action
Nakajima has constructed so-called reflection functors that are isomorphisms between different quiver varieties that have the same framing $\mathbf{w}:$
$$\Phi_{\sigma}:\mathfrak{M}_\zeta(Q,\mathbf{v}...
- 1,687
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91
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Hopf algebras structure and quantum affine algebras
I'm looking for some information about the Hopf algebras structure and the quantum groups.
In particularly I was wondering if (and eventually where) is defined in the case of quantum affine algebras ...
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3
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83
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Composition of operators in $w_{1+\infty}$ and $W_{1+\infty}$
The algebra $W_{1+\infty}$ can be defined as a central extension of the lie algebra $w_{1+\infty}$ (defined as being spanned by $\left(-\partial_z \right)^m z^{-k}$ ). See for example: Alexandrov, ...
user35360
3
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105
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Existence of a maximal rank CR Lie subalgebra
Let $\mathfrak{g}$ be a real Lie algebra of dimension $2n+1$ and let $\mathfrak h \subset \mathfrak g \otimes \mathbb C$ be a subalgebra of complex dimension $n+1$ satisfying $\mathfrak h + \overline{\...
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Reference request: Category of finite dimensional representations of loop algebra is not semisimple
For $\mathfrak{g}$ a semisimple Lie algebra, we may define its (untwisted) loop algebra as $L(\mathfrak{g}) = \mathfrak{g} \otimes \mathbb{C}\lbrack t,t^{-1} \rbrack$. Let $\mathcal{F}$ be the ...
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3
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232
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Lifting a Frobenius endomorphism under an étale morphism
Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
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3
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93
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Understanding the geometric fibre twisted differential operators
Let $\mathfrak{g}$ be a Complex semisimple Lie algebra with decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}$. For $\lambda \in \mathfrak{h}^*$, we let $\mathcal{D}_{\lambda}$ be ...
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3
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88
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Embedding of Verma modules in Kac-Moody Lie algebras
Let $\mathfrak{g}(A)$ be a symmetrizable Kac-Moody Lie algebra over $\mathbb{C}$ and ($\mathfrak{h}$, $\Pi, \Pi^\vee)$ be a realization of the GCM $A$. Assume that $$\mathfrak{g}(A)=\mathfrak{h} \...
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3
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237
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Baker–Campbell–Hausdorff formula for exponential of general Hermitian operators
Let $A$ and $B$ be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for $e^A e^B$ as $e^C=e^A e^B$, for $C$ in the Lie algebra generated by $...
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3
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130
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Lie subalgebra as the intersection of the subalgebra and the Lie subalgebra
Let $X = (X_1, \dots X_q)$ be indeterminates. Let $A(X)$ be the free algebra over $X$. Let $L(X) \subset A(X)$ be the free Lie algebra over $X$.
I consider some finite set $Y \subset L(X)$ and ...
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3
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321
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Maximal symmetry at the speed of light
Are there examples of 1 + 3 dimensional pseudo-Riemannian manifolds with 6 dimensional isometry group whose orbits are light-like (i.e., the metric restricted to each orbit is degenerate)?
Here is a (...
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3
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143
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What kind of equivalences exist between categories of characteristic $0$ and characteristic $p$?
The tilting equivalence for perfectoid algebras gives an equivalence of categories $$K\text{-perf} \cong K^\flat\text{-perf}$$
where the left-hand-side are algebras in characteristic zero and the ...
- 4,164
3
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95
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A question to the derived length in modular group algebras
Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...
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3
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177
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Lie algebra cohomology of loop algebra
Let $G$ be a simple algebraic group over the complex numbers. Is it true that the Lie algebra cohomology $H^*(L\mathfrak{g}, \mathbb{C})$ of the loop Lie algebra $L\mathfrak{g}=\mathfrak{g} \otimes \...
- 3,019
3
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183
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Origin of the standard result on convex hull of weights of an irreducible finite dimensional representation?
What is the earliest published statement and proof of the well-known result: for a simple Lie algebra over $\mathbb{C}$ or other algebraically closed field of characteristic 0, the convex hull (in the ...
- 52.9k
3
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123
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Decomposition of Schur modules over the orthogonal group
Let $V=\mathbb{R}^n$ and $O(n)$ the orthogonal group acting with its standard action on $V$. Now for any partition $\lambda$ we have the Schur module $S_\lambda V$ which is a representation of $O(n)$. ...
- 3,031
3
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82
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The Lie subgroup corresponding to inner derivations
Let $\mathfrak{g}$ be a finite-dimensional real or complex Lie algebra. We know that $Aut(\mathfrak{g})$ is a closed real or complex Lie subgroup of $GL(\mathfrak{g})$. We also know that the Lie ...
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3
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71
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Centraliser of $\Delta U$ in $U\otimes U$
Let $U$ be a universal enveloping algebra of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. What is a good reference for the centralizer of $\Delta (\mathfrak{g})$
in $U \otimes U$ ? Here $\...
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3
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answers
95
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Lie structure over $R$-module
In Higgins' paper Baer invariants and the Birkhoff-Witt theorem (J. Algebra 11 (1969) 469–482, doi:10.1016/0021-8693(69)90086-6) the following definition is given:
A Lie structure over the $R$-module ...
- 427
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181
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Description of real roots of Kac—Moody algebra
Let $\Delta$ be a root system associated to a generalized Cartan matrix, $\alpha_1,\ldots,\alpha_n$ its simple roots.
It is known that if $\Delta$ is of finite, affine or hyperbolic type, $\alpha=\...
- 3,186
3
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answers
235
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Moduli space of nilpotent Lie algebras
Fix a nilpotent Lie algebra $L$ over some char 0 field $k$ which is naturally graded, i. e. isomorphic to graded algebra $\bar L$ associated to lower central filtration.
I'm interested in some ...
- 4,599
3
votes
0
answers
255
views
Structure of a group acting on a Hilbert space
Assume a group $G$ acts faithfully by isometries on a separable infinite dimensional Hilbert space $H$ in such a way that the orbits are closed and the quotient $H/G$ is isometric to a finite ...
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3
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57
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Question about Ext group in $\mathcal{O}^\mathfrak{p}$?
Let $W$ be a Weyl group, let $\mu$ be an antidominant weight.
Let $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$. Denote ${}^IW$ the set of minimal length coset ...
- 1,875
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71
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Deformations of nilpotent parts of superalgebras
I have two questions concerning some results in the article "Deformations of nilpotent parts of superalgebras" of N. van den Hijligenberg, J.Math.Phys. 35, 1427 (1994); doi:10.1063/1.530598
After ...
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122
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It there a nice way to describe the structure of Malcev-complete groups?
Let $\mathbb k$ be a field of characteristic zero. The grouplike functor $\mathbb G$ from complete Hopf algebras to groups is a faithful functor. Its image is the category of Malcev-complete groups ...
- 273
3
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answers
106
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Induced $(\mathfrak{g},K)$-modules
Let $G$ be a noncompact simple Lie group, and $G'$ a noncompact reductive subgroup of $G$. Fix a maximal compact subgroup $K$ of $G$ such that the intersection $K'=K\cap G'$ is a maximal compact ...
- 951
3
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112
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Relationship bewteen Kazhdan-Lusztig Vogan polynomial and Kazhdan-Lusztig polynomial
Let $W^I=\{w\in W: w^{-1}\Phi_I^+\subseteq \Phi^+\}$, $W_I$ be the Weyl group generated by $I$ and $w_I$ be the longest element in $W_I$
Let $M(\lambda)$ be the Verma module with highest weight $\...
- 1,875
3
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answers
135
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How to understand extremal vector?
Extremal vectors are defined in Kashiwara's paper. The definition is as follows.
Simple reflections in the Weyl group of $\mathfrak{g}$ acts on the crystal basis of integrable $U_q(\mathfrak{g})$-...
- 6,211
3
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answers
156
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Which kind of functors preserve the bar-construction?
Let C, D be monoidal infinity categories that admit geometric realizations.
Let $F: C \to D$ be a monoidal functor and A an augmented associative algebra of C.
Denote $Bar(A)= \mathbb{1} \otimes_A \...
- 1,361
3
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answers
97
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Reference Request: Branching Rules of $\mathfrak{s}\mathfrak{l}_n$ in $\mathfrak{s}\mathfrak{l}_{n+1}$
I have heard that the branching rules are well-known for the simple Lie algebra $\mathfrak{s}\mathfrak{l}_n$ in $\mathfrak{s}\mathfrak{l}_{n+1}$ over fields of characteristic zero. Where can I find a ...
user117370
3
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0
answers
156
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Questions about ``$p$-canonical basis" for $\widehat{\mathfrak{sl}_n}$ module (wedge power of natural representation)
Let $p$ be a prime number. Consider the natural representation of the affine Lie algebra $\widehat{\mathfrak{sl}_p}$, defined as follows.
$$A = \bigoplus_{i=1}^N \mathbb{C}a_i; \qquad \text{nat}_p = A ...
- 2,216
3
votes
0
answers
78
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Noncompact dual of $\mathrm{Spin}(2n)$ corresponding to $\mathfrak{so}^*(2n)$
Let $U=\mathrm{Spin}(2n)$, which is a simply connected compact simple Lie group, and let $\mathfrak{u}_0=\mathfrak{so}(2n)$, the Lie algebra of $U$. If $\mathfrak{g}_0$ is a noncompact dual of $\...
- 951
3
votes
0
answers
112
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Indecomposablity in purely inseparable extensions
Let $k$ be a field of characteristic $p$ (e.g. the separable closure of $\mathbb{F}_p(t)$) and consider the extension $k(x)/k$ where $x^p\in k$ but $x$ does not. Consider a (finitely generated) ...
- 31
3
votes
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answers
39
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Second order signature operator in diffeomorphism invariant geometry as an image under right regular representation
I would like to understand the following statement taken from this paper, dealing with the so called Transverse Index Theory or in other words with the index theory for diffeomorphism invariant ...
- 9,340
3
votes
0
answers
91
views
Number of vectors such that the projection is decomposable
Let $V$ be a vector space of dimension $n\geq 6$ over the finite field $\mathbb{F}_q$. Let $\omega\in\bigwedge^{n-3}V$ be a nonzero element. Define the annihilator subspace of $\omega$ by $V_\omega=\{...
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