Product expansions of q-character polynomials
The ring of q -character polynomials is a q -analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain class functions defined simultaneously on the groups Gl n ( F q ) for all n , which we also interpret as statistics on matrices. Here, we evaluat...
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| Vydané v: | Journal of algebraic combinatorics Ročník 57; číslo 3; s. 975 - 1005 |
|---|---|
| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
New York
Springer US
01.05.2023
Springer Nature B.V |
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| ISSN: | 0925-9899, 1572-9192 |
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| Abstract | The ring of
q
-character polynomials is a
q
-analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain class functions defined simultaneously on the groups
Gl
n
(
F
q
)
for all
n
, which we also interpret as statistics on matrices. Here, we evaluate these statistics on all matrices and work toward computing the structure constants of the product in this ring. We show that the statistics are periodically polynomial in
q
and governed by universal polynomials
P
λ
,
μ
(
q
)
which we compute explicitly, indexed by pairs of integer partitions. The product structure is similarly polynomial in
q
in many cases, governed by polynomials
R
λ
,
μ
ν
(
q
)
indexed by triples of partitions, which we compute in some cases. Our calculations seem to exhibit several unexpected patterns. Mainly, we conjecture that certain indecomposable statistics generate the whole ring and indeed prove this for statistics associated with matrices consisting of up to 2 Jordan blocks. Furthermore, the coefficients we compute exhibit surprising stability phenomena, which in turn reflect stabilizations of joint moments as well as multiplicities in the irreducible decomposition of tensor products of representations of
Gl
n
(
F
q
)
for
n
≫
1
. We use this stabilization to compute the correlation of the number of unipotent Jordan blocks of two sizes. |
|---|---|
| AbstractList | The ring of q-character polynomials is a q-analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain class functions defined simultaneously on the groups Gln(Fq) for all n, which we also interpret as statistics on matrices. Here, we evaluate these statistics on all matrices and work toward computing the structure constants of the product in this ring. We show that the statistics are periodically polynomial in q and governed by universal polynomials Pλ,μ(q) which we compute explicitly, indexed by pairs of integer partitions. The product structure is similarly polynomial in q in many cases, governed by polynomials Rλ,μν(q) indexed by triples of partitions, which we compute in some cases. Our calculations seem to exhibit several unexpected patterns. Mainly, we conjecture that certain indecomposable statistics generate the whole ring and indeed prove this for statistics associated with matrices consisting of up to 2 Jordan blocks. Furthermore, the coefficients we compute exhibit surprising stability phenomena, which in turn reflect stabilizations of joint moments as well as multiplicities in the irreducible decomposition of tensor products of representations of Gln(Fq) for n≫1. We use this stabilization to compute the correlation of the number of unipotent Jordan blocks of two sizes. The ring of q -character polynomials is a q -analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain class functions defined simultaneously on the groups Gl n ( F q ) for all n , which we also interpret as statistics on matrices. Here, we evaluate these statistics on all matrices and work toward computing the structure constants of the product in this ring. We show that the statistics are periodically polynomial in q and governed by universal polynomials P λ , μ ( q ) which we compute explicitly, indexed by pairs of integer partitions. The product structure is similarly polynomial in q in many cases, governed by polynomials R λ , μ ν ( q ) indexed by triples of partitions, which we compute in some cases. Our calculations seem to exhibit several unexpected patterns. Mainly, we conjecture that certain indecomposable statistics generate the whole ring and indeed prove this for statistics associated with matrices consisting of up to 2 Jordan blocks. Furthermore, the coefficients we compute exhibit surprising stability phenomena, which in turn reflect stabilizations of joint moments as well as multiplicities in the irreducible decomposition of tensor products of representations of Gl n ( F q ) for n ≫ 1 . We use this stabilization to compute the correlation of the number of unipotent Jordan blocks of two sizes. |
| Author | Balachandran, Adithya Huang, Andrew Sun, Siwen Gadish, Nir |
| Author_xml | – sequence: 1 givenname: Adithya surname: Balachandran fullname: Balachandran, Adithya organization: Massachusetts Institute of Technology – sequence: 2 givenname: Nir orcidid: 0000-0003-4479-0537 surname: Gadish fullname: Gadish, Nir email: gadish@umich.edu organization: University of Michigan – sequence: 3 givenname: Andrew surname: Huang fullname: Huang, Andrew organization: Massachusetts Institute of Technology – sequence: 4 givenname: Siwen surname: Sun fullname: Sun, Siwen organization: Harvard University |
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| Cites_doi | 10.1016/j.aim.2019.04.026 10.1007/s00026-016-0336-7 10.1016/j.ejc.2016.05.004 10.1090/proc/14781 10.1093/imrn/rny144 10.1016/j.jalgebra.2017.03.010 10.1215/00127094-3120274 |
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| DOI | 10.1007/s10801-022-01208-0 |
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| References | N Gadish (1208_CR7) 2019; 148 JB Lewis (1208_CR9) 2016; 58 T Church (1208_CR2) 2015; 164 1208_CR4 1208_CR3 1208_CR1 1208_CR8 J Wan (1208_CR10) 2019; 349 N Gadish (1208_CR6) 2017; 480 J Fulman (1208_CR5) 2016; 20 |
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| Snippet | The ring of
q
-character polynomials is a
q
-analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain... The ring of q-character polynomials is a q-analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain class... |
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| Title | Product expansions of q-character polynomials |
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