Timeline for Symmetric polynoms are Hopf algebra ? What for one needs co-product ?
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| Jul 13, 2020 at 13:44 | comment | added | Clemens Koppensteiner | I may be confused, but the coproduct in terms of power sums seems incorrect to me. Shouldn't $\Delta(p_ip_j) = 1 \otimes p_ip_j + p_i \otimes p_j + p_j \otimes p_i + p_ip_j\otimes 1$? | |
| Jan 19, 2012 at 12:01 | comment | added | darij grinberg | ... function $s_{\lambda}$, which is symmetric), or using some very basic invariant theory (prove that a polynomial function in the entries of a matrix which is invariant under conjugation of this matrix must be a symmetric function of its eigenvalues). | |
| Jan 19, 2012 at 12:00 | comment | added | darij grinberg | If you define ch(V) as Dan Petersen does, then yes, because all the $p_i$ are symmetric. If you define ch(V) as "the symmetric function which, applied to the eigenvalues of a matrix, gives the trace of its action on the $\mathrm{GL}_n$-module $\mathrm{Hom}_{S_n}\left(V,X^{\otimes n}\right)$", then it follows from the very definition, but you need to check that there exists such a symmetric function. This can be done either using the representation theory of $S_n$ (check that all irreducible representations come from partitions, and $\mathrm{ch}\left(V_{\lambda}\right)$ is the Schur ... | |
| Jan 19, 2012 at 6:49 | comment | added | Alexander Chervov | Is it easy to see that ch(V) - symmetric polynom ? | |
| Jan 18, 2012 at 15:25 | comment | added | Dan Petersen | Anyway, the isomorphism between representations and symmetric functions comes now from the fact that the invariants under the general linear group that one can associate to a matrix $A$ are exactly the symmetric functions of its eigenvalues. The particular shape of the formula is then explained by Schur-Weyl duality and that the trace of $A^n$ is exactly the power sum $p_n$ of the eigenvalues of $A$, but I do not remember the details ... this should be in many books though. | |
| Jan 18, 2012 at 15:24 | comment | added | Dan Petersen | Secondly, I do not claim to really understand this isomorphism. But I believe it is most naturally thought of in two steps, first by identifying representations of $S_n$ and $GL_n$, and then representations of $GL_n$ and symmetric functions. Lots of things become nicer by thinking in terms of $GL_n$ (or rather $\mathfrak{gl}_\infty$, as we have infinitely many variables), for instance Young diagrams become highest weight vectors, and the determinantal formula for Schur polynomials becomes the Weyl character formula. | |
| Jan 18, 2012 at 15:23 | comment | added | Dan Petersen | One could write a book answering these questions! Let me just shortly say this: the reason I find this theory useful is as a tool to prove things about representations of $S_n$. You want to understand something about representations, then you translate the question into symmetric functions, do the (suddenly very concrete) computations there, then translate back. For instance the mysterious plethysm of representations becomes a simple algorithmic procedure on the side of symmetric functions. | |
| Jan 18, 2012 at 14:15 | comment | added | Alexander Chervov | What means "usual inner product on symmetric functions" ? | |
| Jan 18, 2012 at 14:09 | comment | added | Alexander Chervov | What means "graded piece $\Lambda^n$" ? Is $p_k$ in formula 4 (definition of $\Psi$) is also a power sum ? If yes how one can come to such a definition of characteristic map - I mean why is it natural? What is it used for ? | |
| Jan 18, 2012 at 13:51 | comment | added | Dan Petersen | Thanks for the correction Darij. I added the definition of how you go from representations of the symmetric group to symmetric functions. | |
| Jan 18, 2012 at 13:49 | history | edited | Dan Petersen | CC BY-SA 3.0 |
added 628 characters in body
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| Jan 18, 2012 at 13:39 | comment | added | darij grinberg | The reason I prefer not to define the operations of $\mathbf{Symm}$ in terms of power sums is that power sums don't generate $\mathbf{Symm}$ if $k=\mathbb Z$ (but only if $k$ is a $\mathbb Q$-algebra). This is probably not particularly relevant for Alexander, though. | |
| Jan 18, 2012 at 13:38 | comment | added | Alexander Chervov | Thank You very much ! But would you be so kind to explain how reps of S_n, S_k are related to symmetric functions ? The second definition is pretty explicit (so I like it), but un-motivated :( And I added - question: "where is it useful" ? | |
| Jan 18, 2012 at 13:10 | history | answered | Dan Petersen | CC BY-SA 3.0 |