Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [symmetric-algebra]

A quotient of the tensor algebra by the ideal generated by symmetric tensors.

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Analogue of the Koszul complex for symmetric powers?

Let $R$ be a commutative ring and let $M$ be a free $R$-module of rank $n$. Let $s: M \rightarrow R$ be a $R$-linear map. Then, the Koszul complex is a complex $0 \rightarrow \bigwedge^{n}M \rightarrow \bigwedge^{n-1}M \rightarrow \dots…
Sunny Sood
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What are the algebraic features and the geometric interpretation of the symmetric algebra?

The exterior algebra is a quotient of the tensor algebra that gives an anti symmetric product. The symmetric algebra is similar except that the product is symmetric. The objects in both algebras are not tensors. I know that the exterior algebra…
Bible Bot
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Determinant of exterior power and symmetric power

I am self studying the tensor algebra, symmetric algebra and exterior algebra. The exterior algebra brings a new definition of determinant to me. I am curious what if $f:V \to V$ is a linear endomorphism on $n$ dimensional vector space $V$also with…
Bigdragon
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What is the subspace of alternating (not antisymmetric) tensors?

Let $k$ be a field, and $V$ a finite-dimensional vector space over $n$. We may define some quotient spaces of $V^{\otimes n}$, by stating their universal property in terms of factoring multilinear maps: The symmetric quotient $S^n(V)$, defined by…
Joppy
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Definition of symmetric power of a linear represenation

I'm reading Kowalski's Representation theory, and there's a part about the symmetric and antisymmetric powers of a representation, and I'd like to ask a question about those. So there's a proposition in the book that says if we have a representation…
Matija Sreckovic
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The symmetric algebra of a vector space is generated by powers

Let $V$ be a finite-dimensional vector space over a field $k \supseteq \mathbb Q$ (characterstic $0$). In Helgason's Groups and Geometric Analysis it is mentioned that the symmetric algebra $S(V)$ is linearly generated by the $v^m$ for $v \in V$ and…
Bart Michels
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Mixed symmetric algebra and applications to linear algebra?

For dual finite-dimensional vector spaces $V,V^*$ the "mixed exterior algebra" $$\textstyle\bigwedge(V^*,V)=\bigwedge V^*\otimes\bigwedge V$$ is a powerful tool for studying linear transformations of $V$ (see here for a brief intro and example).…
blargoner
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$ G $-invariant symmetric bilinear map and trivial subrepresentation of symmetric square

Let $\pi:G\rightarrow \text{GL}(V)$ be a finite-dimnesional complex representation of finite group $ G $. If there is a non-zero $ G $-invariant symmetric bilinear form on $ V $, then the symmetric square $\text{Sym}^2(V)$ must contain the trivial…
Kevin
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Is Fock space a symmetric/exterior algebra?

Wikipedia's Fock Space entry says that the Fock space is the direct sum of tensor products of $H$. However, it is not represented as $\bigoplus_{n=0}^{\infty} H^{\otimes n}$, but rather as $\bigoplus_{n=0}^{\infty} S_\nu H^{\otimes n}$, where…
Godfly666
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Connections between Weyl, Clifford, Exterior, and Symmetric algebras

The Clifford algebras are analogous to the Weyl algebras in the same way that the exterior algebras are analogous to the symmetric algebras. How are the Clifford and Weyl algebras analogous? The Clifford algebras are about generators that square to…
Lave Cave
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Harish Chandra isomorphism:Invariant polynomial functions

I am trying to read the complete proof of Harish Chandra Isomorphism theorem from the book of Humphreys. Notations: $L$ is a finite dimensional semisimple Lie algebra with Cartan subalgebra $H$. $G$ is the inner automorphism group of $L$ i.e. the…
Irfan
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Symmetric square of the fundamental representation of $SU(2)$ is irreducible

I am trying to prove that the symmetric square of the fundamental representation $V$ of $SU(2)$ is irreducible. Here is my approach: Using the famous character formula of the symmetric representation and knowing that the fundamental representation…
F.H.A
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Proving Symmetric Difference of A and B

Let A and B be sets. Define the symmetric difference of A and B as A∆B= (A ∪ B) − (A ∩ B). (a) Prove that A∆B = (A − B) ∪ (B − A) I tried to start this but am getting really lost. if someone could try to help that would be great
Sascha816
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Bilinear form on a finite dimensional algebra over a field

I am reading the first chapter of 'methods of representation theory, Volume I' section 9A, written by Charles W.Curtis & Irving Reiner. Let $A$ denotes a finite dimensional algebra over an arbitrary field $K$ and let $\beta : A \times A \rightarrow…
Oe-thau
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Representation of a finite group $G$ on the symmetric and exterior power of a $\mathbb C G$-module

Suppose $G$ is a finite group, and $V$ is a $\mathbb C G$-module. We can equip $V\otimes V$ with the structure of a $\mathbb C G$-module via $$g(v\otimes w) = gv\otimes gw.$$ I am trying to understand how this defines a $\mathbb C G$-module…
algibberishian
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