# Questions tagged [symmetric-algebras]

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### Almost split sequences for symmetric algebras

Let $k$ be an algebraically closed field and $A$ be a symmetric algebra.
I want to know how to compute almost split sequences ending at a non-projective indecomposable right $A$-module $X$.
Question: ...

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### Symmetric algebra of monomial ideal modulo a binomial ideal isomorphic to a quotient of a polynomial ring

Note: I posted this question on MSE but haven't received any response.
I’m trying to understand the proof of Corollary 1.9 in “Binomial ideals” by David Eisenbud and Bernd Sturmfels.
Notation: Let $S= ...

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### Formula for the matrix units in the Gelfand-Tsetlin basis of the symmetric group algebra?

Are there any formulas for the irreducible off-diagonal elements $E^{\lambda}_{ij}$ in the Gelfand-Tsetlin basis of the symmetric group algebra $\mathbb{C}[S_n]$?
Here is the context for my question. ...

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### Symmetric algebra over a realization of Coxeter System is a dgg algebra

I have been reading a paper of Achar, Makisumi, Riche and Williamson. In the chapter 3, the authors talk us of bigraded modules and dgg modules and I'm stuck here.
Let $(W, S)$ be a Coxeter system, ...

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### What comes next in the sequence "symmetric algebras, exterior algebras, divided power algebras, ..."?

This question was posed by A Rock and a Hard Place in this discussion, where they mentioned the isomorphisms
\begin{align*}
\mathrm{L}\,\mathrm{Sym}^n_R(M[1]) &\cong (\mathrm{L}\,{\...

2
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### The shuffle algebra over the rationals is isomorphic to the polynomial algebra in the Lyndon words

On this wikipedia page is stated that over the rational numbers, the shuffle algebra (over a set $X$) is isomorphic to the polynomial algebra in the Lyndon words (on $X$). I was wondering if you can ...

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### What is the definition of this function?

I'm reading a paper and I didn't understand this notation used by the author:
Let E be a vector space and F be a subspace of E.
Let $S(E/F)$ be the symmetric algebra of $E/F$. For every element $P \...

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### Inequality for trace of a symmetric product?

Let $A$ be a real, positive-definite, symmetric operator on an $n$-dimensional space $V$. Write $\odot^k A$ for the action of $A$ on the symmetric power $\odot^k V$. Let $v_1,\dotsc,v_n$ be a basis ...

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### Is the (super-)symmetric power of the exterior algebra free?

Let $V$ be a vector space over $k$ of dimension $m$. (I'm only interested in the case $k=\mathbb{Q}$.) Let $R:=\Lambda^*V$ be the exterior algebra. It carries the structure of a supercommutative ring: ...

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### Symmetric algebras of given dimension

Fix an algebraically closed field $F$. Are there only finitely many symmetric algebras with unit over $F$ of a given finite dimension (up to isomorphism)? By symmetric I mean a Frobenius algebra where ...

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### Symmetric power of an algebra

Given an algebra $A$ over $k$ with characteristic zero and a positive integer $n$, the subspace of $A^{\otimes n}$ consisting of all tensors invariant under the action of all permutations $\sigma\in\...

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### A nice generating set for the symmetric power of an algebra

I'm looking for a reference for the following fact.
Suppose $A$ is a finitely generated associative commutative unital algebra over an algebraically closed field of characteristic zero. Let $S^n(A)$ ...

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### Coderivations of S(V) correspond to linear maps S(V) -> V. Only over characteristic 0?

Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
$\Delta ...

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### Restricted universal enveloping algebra of Abelian p-Lie algebra

Question: Let $p$ be a prime. Let $k$ be a commutative ring such that $p=0$ in $k$.
Let $\mathfrak g$ be an abelian $p$-restricted Lie algebra over $k$. In other words, let $\mathfrak g$ be a $k$-...

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### Where is there a treatment of "exponential monads"?

I have a category $C$, which is equipped with a symmetric monoidal structure (tensor product $\otimes$, unit object $1$). My category also has finite coproducts (I'll write them using $\oplus$, and ...

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### Are cyclotomic Khovanov-Lauda-Rouquier algebras symmetric?

Recall that for k a field, a finite dimensional k-algebra A is called symmetric if it is isomorphic to its dual as a bimodule of itself. Which is to say, there's a trace map t:A -> k such that t(...