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If I remember correctly, one of the symmetric powers should become divided powers. A quick google search brought me to Section 3.4 of Aprodu, Farkas, Papadima, Raicu, and Weyman - Koszul modules and G …

I think this might be so standard that there is no obvious reference. The cohomology has a basis of Schubert classes. In the adjoint case, Schubert classes are indexed by the Weyl group orbit of the h …

There are lots of papers dealing with representation-theoretic questions and universal enveloping algebras using Gröbner bases. Some examples are given by these: 1, 2, 3, 4.

I am interested in the dimension, or, even better, in the $S_8$-module structure of the space of degree 8 multilinear operations on Jordan algebras. Recall that a Jordan algebra is a commutative but n …

I managed to run Albert on a very powerful computer at work, and the computation of the desired dimension converged: it seems equal to 19089. I would very much like to confirm that it is correct (I am …

There is an instance where the two formulas are very close: if $f_1,\ldots,f_n$ are fundamental solutions of a linear ODE with indeterminate (constant) coefficients. This is explained in the elegant p …

I accidentally (looking for something else) came across another paper where a very elegant explanation is given: Dan Laksov, Anders Thorup: Schubert Calculus on Grassmannians and Exterior Powers India …

This is very standard. For a compact complex variety admitting a cell decomposition, the (co)homology is the free Abelian group generated by the cells (over $\mathbb{C}$ there is no room for the diffe …

In your context, you want to think of the Schur-Weyl duality as a way to construct representations of $GL(V)$ out of representations of symmetric groups. To give a precise answer along these lines tha …

The method/result you are looking for is commonly known under the name "unitary trick" (of Hurwitz and Weyl), - this keyword should bring you a great deal of accessible explanations of all possible le …

The Whitehouse module referred to in one of the other answers is not necessary, since it is related to the cyclic operad Lie, that is to the representation of $S_{n+1}$ in $Lie(n)$. The decomposition …

I think that there is just a little mess between things that are denoted $K$, $K'$ in the book, as well as $d$, $d'$. For that, let us examine these formulas carefully. Using the first formula for the …

As requested, I elaborate on my comment. First of all, let me make a change of variables $a_i=U_i-1$. The relations then become $a_i+1=a_{i-1}a_{i+1}$. For $n=2,3,4$ I used the Magma online calcula …

There are two very famous instances of Jack polynomials in relationship to the Virasoro algebra (there are some others, but they very often seem to be related to one of these): Katsuhisa Mimachi and …

Yes. See the result of Section 2.5 of a wonderful paper of Bernhard Keller : https://webusers.imj-prg.fr/~bernhard.keller/publ/ilc.pdf (and the references therein).