## Schur functors generalization to “Jack”, “Hall-Littlewood”, “Macdonald” functors ?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

Schur functors are functors from the category of vector spaces to itself. If we take an operator $M: V->V$ and apply a Schur functor to it and then calculate trace $Tr(M^{\Lambda})$ we will get Schur polynomial in the eigenvalues of $M$.

Question Can one generalize (deform) Schur functors, such that $Tr(M^{\Lambda})$ will give polynomials which generalize (deform) Schur polynomials e.g. Hall-Littlewood polynomial, or Jack polynomial and most generally Macdonald polynomials ?

-

1
Have you seen arxiv.org/abs/q-alg/9503012 ? – Gjergji Zaimi Jun 4 at 9:05
@Gjergii Thank you I know, may be I forget something now, but it seems to me it is not the answer. Why do I need intertwiner? is it natural ? It does seems to me so. Morever you will need to take very specific representation to obtain Calogero model (which corresponds to Jack polynoms, respectively in q-case to Macdonalds)... – Alexander Chervov Jun 4 at 14:14

## 1 Answer

It seems to me that this is answered, perhaps in a boring way, by Haiman's work on the $n!$-conjecture (now a theorem due to Haiman). For any partition $\lambda$, Haiman constructs a finite dimensional graded module $C_{\lambda}$ for $\mathbb{C}[x,y][S_n]$. (The group elements commute with $x$ and $y$, and $x$ and $y$ commute with each other.) The doubly graded Frobenius character of $C_{\lambda}$ is the $\lambda$-Macdonald polynomial.

Now just use Schur-Weyl duality: Define the functor $F_{\lambda}$ from vector spaces to vector spaces by $$V \mapsto V^{\otimes |\lambda|} \otimes_{\mathbb{C}[S_n]} C_{\lambda}.$$ The result is a doubly graded $\mathbb{C}[x,y]$ module which is the sum of Schur functors corresponding to the Macdonald polynomial.

-

@David thank you very much ! It is unexpected and interesting form me ! (What is "Frobenius character") Actually I hoped something like a deformation e.g. vector spaces will be the same but morphisms somehow deformed. What do you think is it possible have something like this ? Morally this should correspond to quantum groups - take V as basic representation of U_q(gl). Send "V" to irrep of U_q(gl) which corresponds to Young diagram Lambda. But it is not clear for me how to make functor from this... – Alexander Chervov Jun 4 at 14:23
The "Frobenius character" is the standard map which sends an $S_n$-representation to a symmetric polynomial. The funny thing is that it is NOT a character -- in particular, the Frobenius character of a tensor product is not the product of the Frobenius characters. The relationship goes through Schur-Weyl duality: If $M$ is an $S_n$-rep with Frob. character $f$, then $V^{\otimes n} \otimes_{k[S_n]} M$, as a $GL(V)$-rep, has character $f$. – David Speyer Jun 4 at 15:10
@David is it the same thing which is called "characteristic map" ? arxiv.org/abs/1112.0620 Characteristic maps for the Brauer algebra A. I. Molev, N. Rozhkovskaya – Alexander Chervov Jun 4 at 16:18
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 23, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8999674916267395, "perplexity_flag": "middle"}