Questions tagged [exterior-algebra]

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22 votes
4 answers
6k views

"Natural" pairings between exterior powers of a vector space and its dual

Let $V$ be a finite-dimensional vector space over a field $k$, $v_1, \dotsc v_n \in V$ a set of vectors, and $f_1, \dotsc f_n \in V^{\ast}$ a set of covectors. Up to permutation, there seem to be at ...
Qiaochu Yuan's user avatar
52 votes
9 answers
12k views

Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
agt's user avatar
  • 4,246
19 votes
3 answers
941 views

How big can a wedge of $n$ 2-forms in $\mathbb{R}^{2n}$ be?

$\def\RR{\mathbb{R}}$Let $\omega$ be a $2$-form on $\RR^{2n}$, where $\RR^{2n}$ has the usual Euclidean norm. The comass of $\omega$ is defined to be $\max_{|u|, |v| \leq 1} \omega(u,v)$. Here is the ...
Mikhail Katz's user avatar
  • 15.1k
15 votes
2 answers
2k views

Clifford PBW theorem for quadratic form

$\DeclareMathOperator\Cl{Cl}$Update Feb 3 '12: now with a question 2 which is much more elementary (and should be well-known!). Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-module, and $g:...
darij grinberg's user avatar
13 votes
3 answers
682 views

Ring of invariants of $\operatorname{SL}_6$ acting on $\Lambda^3 \mathbb C^6$

Let $G=\operatorname{SL}_6$ act on $V=\Lambda^3 \mathbb C^6$. I would like to find the ring of invariants $\mathbb C[V]^G$. There is an obvious invariant $$Sq: V \to \mathbb C, \quad \omega \mapsto \...
evgeny's user avatar
  • 1,980
12 votes
1 answer
401 views

Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?

Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor: $$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ...
Asaf Shachar's user avatar
  • 6,611
8 votes
1 answer
717 views

Deligne's exterior power

In "Catégories Tannakiennes", Deligne defines the $n$th exterior power of an object $A$ of an abelian tensor category $\mathcal{C}$ as the image of the morphism $$p : A^{\otimes n} \to A^{\otimes n}, ...
Martin Brandenburg's user avatar
28 votes
1 answer
2k views

Analogy between the exterior power and the power set

The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$. The usual definition of the exterior ...
Martin Brandenburg's user avatar
19 votes
2 answers
8k views

Distributing the Hodge map over the wedge product

Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to "...
user49105's user avatar
  • 191
17 votes
1 answer
634 views

An explicit reconstruction of a matrix from its minors

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ $\newcommand{\Cof}{\operatorname{cof}}$ Let $V$ be a $d$-dimensional real vector space. ($d \ge 4$). Fix an odd integer $...
Asaf Shachar's user avatar
  • 6,611
10 votes
1 answer
2k views

Exterior powers in tensor categories

Let $\mathcal{C}$ be a cocomplete $R$-linear tensor category. Many notions of commutative algebra can be internalized to $\mathcal{C}$. For example a commutative algebra is an object $A$ in $\mathcal{...
Martin Brandenburg's user avatar
8 votes
2 answers
444 views

Obstructions for the wedge of coordinate differentials to be harmonic

Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property: For every $p \in M$ there exist a ...
Asaf Shachar's user avatar
  • 6,611
7 votes
1 answer
412 views

How big can a wedge of 2-forms be?

The comass of a 2-form $\alpha$ is the maximal value of $\alpha(u,v)$ for a pair of unit vectors $u,v$. The symplectic form $\alpha$ on $\mathbb R^{2n}$ has the property that $|\alpha^{\wedge n}| = n!...
Mikhail Katz's user avatar
  • 15.1k
7 votes
2 answers
1k views

When is the exterior algebra a Hopf algebra?

I have several questions on the exterior algebra of a vector space: Q1:When has the exterior algebra A (viewed just as an algebra, not considered as a graded algebra) of an $n$-dimensional vector ...
Mare's user avatar
  • 25.8k
6 votes
0 answers
254 views

Is a Sobolev map with invertible smooth minors smooth?

$\newcommand{\Cof}{\text{cof}}$ Let $k,d$ be even integers, such that $d\ge3$ and $2 \le k \le d-1$. Let $\Omega \subseteq \mathbb{R}^d$ be open, and let $f \in W^{1,p}(\Omega,\mathbb{R}^d)$, for ...
Asaf Shachar's user avatar
  • 6,611
6 votes
1 answer
1k views

What is the role of topology on infinite dimensional exterior algebras?

Wedge products and exterior powers are discussed in W. Greub's book Multilinear algebra as follows. Definition: Let $E$ be an arbitrary vector space and $p \ge 2$. Then a vector space $\bigwedge^{p}E$ ...
MathMath's user avatar
  • 1,255
5 votes
3 answers
609 views

An isomorphism of 2-Schur modules

This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings ...
darij grinberg's user avatar
3 votes
3 answers
488 views

Can we specify the value of harmonic forms at a point?

Let $M$ be a smooth $d$-dimensional oriented Riemannian manifold, and let $1 < k < d$ be fixed. Let $p \in M$, and let $\alpha_p \in \bigwedge^k(T_pM)^*$. Does there exist an open ...
Asaf Shachar's user avatar
  • 6,611
3 votes
1 answer
262 views

Grassmannian as a submanifold of $\Lambda^m(E)$.

Let $E$ be a vector space of dimension $d\ge4$ over $K$, and $2\le m\le d$ be an integer. I am interested in the characterization of those elements $\omega$ of $\Lambda^m(E)$ that can be written in ...
Denis Serre's user avatar
  • 51.5k