Questions tagged [exterior-algebra]

It is a quotient - of the tensor algebra, obtained by taking graded sum over whole numbers $n$ of $n$-fold tensor products - by the ideal generated by elements of the form $a\otimes a$. We write the residue class of $a\otimes b$ in this algebra, as $a\wedge b$ and call it the wedge product.

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10
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0answers
94 views

Relationship between exterior power of representation and variance?

I was reading the question: Symmetric and exterior power of representation regarding how to determine the character of an exterior power of a representation from the original representation. One of ...
9
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1answer
426 views

How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace. I ...
9
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621 views

Hodge-Star-Operator on arbitrary oriented basis

Assume that $V$ is oriented finite dimensional vectorspace with dimension $n$, $g \in T^0_2(V)$ a given symmetric and nondegenerate tensor. Let $\mu$ be the corresponding volume element of $V$. ...
8
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51 views

How many matrix minors determine all the minors?

Let $n$ be a positive integer, and let $1<k<n$. Suppose we have an "unknown" real $n \times n$ matrix $A$. (we do not know the entries of $A$). Can we recover all the $k$-minors of $A$ from ...
8
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1answer
102 views

A characterization of the subgroup of $\text{GL}(\bigwedge^k V)$ which preserves pure tensors?

Let $V$ be a $d$-dimensional real vector space, and let $1<k<d$. Set $$H=\{B\in\text{GL}(\bigwedge^k V) \, | \, B \,\text{ preserves pure tensors }\}$$ (i.e. $B \in H$ if it maps ...
8
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1answer
185 views

Which metrics on exterior power are induced from metrics on the base?

$\newcommand{\id}{\text{id}}$ $\newcommand{\Hom}{\text{Hom}}$ Let $V$ be a $d$-dimensional real vector space, and let $2 \le k \le d-1$. Every inner product on $V$ induces an inner product on $\...
8
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577 views

Free Graded Commutative Algebra on a Graded Vector Space

Let $V$ be a graded vector space, thought of as a collection $\{ V^n \}_{n \ge 0}$ of vector spaces. Let $V_{odd} = \bigoplus_{n \text{ odd}} V^n$ and $V_{even} = \bigoplus_{n \text{ even}} V^n$. I am ...
7
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156 views

Is there a linear injection $ \Lambda^k V^* \otimes \Lambda^k V^* \to \Lambda^k (V^* \otimes V^*)$ which preserves decomposability?

Let $V$ be an $n$-dimensional real vector space, and let $2 \le k \le n-2$. Definitions We say an element $\omega \in \Lambda^k V$ is decomposable if $\omega=\alpha_1 \wedge \dots \wedge \alpha_k$, ...
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260 views

What is the relation between the determinant of the adjoint representation, and volume or eigenvalues?

I'm relatively new to Lie Groups, and finding in a text the following statement, as it is supposed to be obvious. I would love some reference or context to either of these two (probably related) ...
7
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206 views

Why is the symbol for the exterior product a meet rather than a join?

I've moved this over to HSM. It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (...
7
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247 views

Frobenius Condition for Singular Integrable Distributions

A smooth "singular" distribution $D\subseteq TM$ on an $n$-dimensional manifold $M$ is integrable if it is tangent to immersed submanifolds $N_\alpha$ that are disjoint and cover $M$. If dim$D=k$ ...
7
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1k views

What is the difference between tensor calculus and exterior derivative type concepts?

I am trying to clarify terms in order to help me figure out what I'd like to study. I understand that $p$-forms and $p$-vectors are used with things like wedge products, exterior algebras, and a ...
7
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796 views

Lie algebra: symmetric and exterior power of representation

If $\mathfrak{g}$ is a Lie algebra, $V$ and $W$ are representation of $\mathfrak{g}$ we define the action of $\mathfrak{g}$ on $V \otimes W$ in the following way: $X \cdot (v \otimes w)=(X \cdot v) \...
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160 views

Clifford algebra is isomorphic to exterior algebra, proof in Lawson's

This Proposition 1.2, pg 10 of Lawson's Spin Geometry. Context: We have a homomorphism of graded algebra $$ \wedge^*(V) \rightarrow G^*$$ induced on each component from the map $$\wedge^r(V) \...
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256 views

Understanding the derivation of this form of the coderivative

Let $M$ be a smooth Riemannian manifold; Let $E$ be a vector bundle over $M$, equipped with a metric and a compatible connection $\nabla$. Denote by $d:\Omega^k(M,E) \to \Omega^{k+1}(M,E)$ the ...
6
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1answer
266 views

What textbook/reference should I read in order to answer these questions?

Might be a strange question, but what textbook/reference should I read in order to be able to solve problems like the followings? I only took one class in classical differential geometry, and we ...
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94 views

Generalization of Liouville's formula to other coefficients of the characteristic polynomial

If $X(t)$ is an $n \times n$ matrix solving linear homogeneous ODE $$ \frac{d}{dt} X(t) = A(t)X(t), $$ then for $\det X(t)$ we have Liouville's formula: $$ \frac{d}{dt} \det X(t) = \text{tr} A(t) \det ...
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202 views

Is this a valid way to think of decomposable $k$-forms?

Let $V$ be a finite dimensional vector space and $\eta \in \Lambda^k(V^\ast)$ be decomposable and non-zero. Then there exists covectors $\omega^1,\dots,\omega^k$ such that $$\eta(v_1, \dots, v_k) = \...
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52 views

When does $v_0\wedge\dots\wedge v_{k-1}=0$ when working over a ring that's not a field?

Let $M$ be a module over a commutative ring $R$, and let $v_0,\dots,v_{k-1}$ be elements of $M$. If $R$ is a field then $v_0\wedge\dots\wedge v_{k-1}$ is equal to $0$ if and only if $v_0,\dots,v_{k-1}$...
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64 views

Abtract explanation for $\det(A+B) = ∑_{k=0}^n \langle \Lambda^k A, \Lambda^{n-k} B \rangle$

Let $A$ and $B$ be $n×n$ matrices. If we expand the determinant of $A+B$ as a sum over all permutations of $[\![1,n]\!]$ and all choices of whether the coefficient comes from $A$ or from $B$, this ...
4
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1answer
203 views

Integration of differential forms - how to extend (and fix?) this intuition?

For context, I'm a first-year undergrad in a linear algebra / multivariable calculus course. I've developed some intuition about $k$-vectors and $k$-forms, and I want to know: Are there any problems ...
4
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47 views

Is $\Lambda$ essentially the unique solution to $F(M)\cong\frac{F(M\oplus R)}{F(M)}$?

Let $R$ be a commutative ring and let $F$ be a functor $\mathbf{Mod}_R\rightarrow \mathbf{Mod}_R$. Then for a module $M$ the split mono $M\rightarrow M\oplus R$ gives a split mono $F(M)\rightarrow F(M\...
4
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598 views

Koszul sign convention and symmetric group action on the graded n-th tensor product

Let $V_\bullet = (V_k)_{k \in \mathbb{Z}}$ and $W_\bullet = (W_k)_{k \in \mathbb{Z}}$ be two graded vector spaces on 0 caracteristic field. We define the tensor product of $V_\bullet$ by $W_\bullet$ ...
4
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1answer
128 views

Exterior powers of vector space and kernel

Let $E$ be a vector space and $A$ a subspace of $E$. Let $q$ be a positive integer. Then we can define a subspace $\Lambda^q A$ of the $q$-th exterior power of $E$ by $$ \Lambda^qA=span\{\ a_1\wedge\...
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102 views

When does the duality functor commute with the wedge power functor?

When working with modules over a fixed commutative ring, I know that $(M \otimes N)^* \cong M^* \otimes N^*$ provided either $M$ or $N$ is finitely generated projective. Does it follow that $(\...
4
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1answer
356 views

Is there an intuitve motivation for the wedge product in differential geometry?

I've recently started studying differential forms and have been looking at differential forms. I'm struggling to understand the motivation for introducing the notion of the wedge product. Does it ...
4
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390 views

Prove that $\phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)].$

I have proved these two exercises: (1) Suppose that $T \in \Lambda^p(V^*)$ and $v_1, \ldots, v_p \in V$ are linearly dependent. Prove that $T(v_1, \ldots, v_p) = 0$ for all $T \in \Lambda^p(V^*)$. ...
4
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1answer
171 views

Computing wedge product of two 1-forms.

Let $L$ be a lattice in $\mathbb{C}$ and let $\pi :\mathbb{C}\to X=\mathbb{C}/L$ be a quotient map. Show that the local formula $dz$ in every chart of $\mathbb{C}/L$ is a well-defined holomorphic 1-...
3
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29 views

Decomposing $k$th exterior powers $\Lambda^kV(\omega_1)$

Let $\Phi$ be a $G_2$ root system, $\omega_1$ the fundamental weight corresponding to the shorter root and consider the unique irreducible highest weight module $V = V(\omega_1)$ of highest weight $\...
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70 views

Does this geometric rigidity condition forces the map to be the identity?

$\newcommand{\Cof}{\operatorname{cof}}$ $\newcommand{\id}{\operatorname{Id}}$ $\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $d$-dimensional vector ...
3
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72 views

Classifying the orbits of the natural $\text{GL}(V)$-action on the exterior power $\bigwedge^k V$

Let $V$ be a real $d$-dimensional vector space, and let $1 < k < d$. Consider the following action of $\text{GL}(V)$ on $\bigwedge^k V$: $(T,\omega) \to (\bigwedge^k T) \omega$. Can we ...
3
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0answers
100 views

Complexified cotangent bundle

I have never worked with differential geometry over $\mathbb{C}$, and I feel a little bit confused. If $(M,J)$ is an almost complex manifold of dimension $2n$, we can extend $J_{p}$ to $T_{p}(M)\...
3
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136 views

Regarding vector-valued (differential) forms

I denote the space of all $V$-valued differential $k$-forms on $M$ with $\mathcal A^k(M,V)$. Let $\omega\in \mathcal A^k(M,V)$ and $\eta\in \mathcal A^l(M,W)$, where $V,W$ are finite real vector ...
3
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0answers
48 views

Translating between two different definitions of exterior derivative

If $\Omega^k(M)$ is the space of differential $1$-forms on a manifold $M$, one may define the operator $\mathop{}\!\mathrm{d} : \Omega^k(M) \to \Omega^{k+1}(M)$ in a coordinate-independent way as ...
3
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1answer
60 views

exterior derivative for parallelizable manifolds

Suppose M has tangent bundle $TM = V \times M$, where $V$ is a vector space. Then all exterior bundles have the form $\wedge^p (T^*M) = \wedge^p V \times M $. In particular, $p$-forms on $M$ are the ...
3
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119 views

How do mathematicians think about multivectors/exterior algebra?

I recently came across the notion of multivectors and exterior algebras recently and while I find these notions to be very enlightening, I also feel a strong temptation to interpret these objects too ...
3
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0answers
169 views

The determinant of $X_I$ = wedge product of elementary tensors

Let $(x_1,\dots,x_k)$ be vectors in $\mathbb{R}^n$; let $X$ be the matrix $X = [x_1 \cdots x_k]$. If $I = (i_1, \dots, i_k)$ is an arbitrary $k$-tuple from the set $\{1,\dots, n\}$, show that $\phi_{...
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150 views

Does the wedge product of bundle-valued forms induce a universal object?

Given a smooth manifold $M$ and a vector bundle $E$ over $M$, the $C^\infty(M)$-module of $E$-valued $p$-forms on $M$ is defined to be $$\Omega^p(M; E) := \Gamma_M\left( \bigwedge^p T^*M \otimes E \...
3
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0answers
220 views

Tensor algebra becomes an R-algebra. Theorem, Proof, Dummit and Foote

I have the definition of tensor algebra as follows: $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$, where $M$ is an $R$ module, where $R$ is commutative and contains the element $1$. Finally $T^k(M) = \...
3
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0answers
899 views

Inner product exterior algebra

I have to prove that if $V$ is a real vector space ($\dim V=n$) with inner product $(.,.)$ then if we define $$ (v_{1}\wedge v_{2}\wedge\cdots\wedge v_{k},w_{1}\wedge w_{2}\wedge\cdots\wedge w_{k}) =...
3
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0answers
421 views

Vector Laplace Beltrami operator on surface tangent and surface normal vector field

Consider a closed, compact, embedded surface $f:M \rightarrow \mathbb{R}^3$ and a vectorfield $X$ on the surface that can be decomposed in the surface frame basis $\{e_1,e_2,e_3\}$, where $\{e_1,e_2\}$...
3
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0answers
193 views

Multiplication in exterior algebra

Take $V = K^{n}$. Let $\omega$ be a non-zero element of $\bigoplus_{k=1}^n \bigwedge^k V$, where we have excluded the summand $\bigwedge^0 V = K$. (1) Prove that there exists an $m > 1$ for which $\...
3
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287 views

Grassmann Algebras

The Grassmann algebra $G$ is the algebra over a field $\mathbb{F}$ generated by the variables $e_i$ such that $e_i^2=0$ and $e_i e_j = - e_j e_i$. I'm looking for some references on algebras $G \...
2
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0answers
28 views

Wedge product for composite of linear transformations

I am studying the wedge product for multivariable analysis, but I feel that the operations are not so intuitive in general. I am looking at the following question which deals with linear ...
2
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0answers
32 views

Is every rotationally-invariant $k$-covector zero?

Let $V$ be a real $n$-dimensional oriented inner product space, and let $1 \le k < n$. I am trying to find different simple proofs for the following claim: There is no non-zero $\omega \in \...
2
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0answers
61 views

Inner product of $k$-forms

I'm working on the following problem from Lee's Introduction to Riemannian Manifolds: Let $(M,g)$ be a Riemannian $n$-manifold. show that for each $k=1,\ldots, n$, there is a unique fiber ...
2
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0answers
42 views

exterior algebra of representations of $S_n$

$$ \begin{aligned} \Lambda^{r} \mathbb{C}^{n} &=\bigoplus_{i=0}^{r}\left(\Lambda^{(r-i)} V \otimes \Lambda^{i} U\right) \\ &=\left(\Lambda^{r} V \otimes U\right) \oplus\left(\Lambda^{r-1} V \...
2
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0answers
85 views

A way to define the wedge product

There are various equivalent ways of introducing the wedge product. One possibility is to first define $\,\wedge\,$ for basis forms, $$ e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_r}\;\equiv\;r!\,\left[\,e^...
2
votes
0answers
52 views

Clarifying a step in an answer about exterior products of coherent sheaves

The question is about the accepted answer here. I decided not to ask in a comment there since the original asker is no longer active. In "Step 2" of the answer given there, Roland claims, "This is ...
2
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0answers
30 views

Finding the Wedge Product of Two Multivariable Vectors and Making Sense of It

I was attempting to solve: $$ -2dx_{1} \land dx_{4}\left( \begin{bmatrix} 2 \\ 3 \\ -5 \\ 2 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \\ 4 \\ -5 \end{bmatrix}\right) $$ I solved this by pulling out the ...