Averages over classical Lie groups, twisted by characters
We compute E<sub><em>G</em></sub>(∏<sub><em>i</em></sub>tr(g<sup>λ<sub><em>i</em></sub></sup>)), where <em>g</em> ∈ <em>G</em> = Sp (2<em>n</em>) or SO (<em>m</em>) (&l...
| Main Author: | |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Elsevier
2007
|
| _version_ | 1826260061861707776 |
|---|---|
| author | Dehaye, P |
| author_facet | Dehaye, P |
| author_sort | Dehaye, P |
| collection | OXFORD |
| description | We compute E<sub><em>G</em></sub>(∏<sub><em>i</em></sub>tr(g<sup>λ<sub><em>i</em></sub></sup>)), where <em>g</em> ∈ <em>G</em> = Sp (2<em>n</em>) or SO (<em>m</em>) (<em>m</em> = 2<em>n</em>, 2<em>n</em> + 1) with Haar measure. This was first obtained by Diaconis and Shahshahani [Persi Diaconis, Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994) 49-62. Studies in applied probability], but our proof is more self-contained and gives a combinatorial description for the answer. We also consider how averages of general symmetric functions E<sub><em>G</em></sub>Φ<sub><em>n</em></sub> are affected when we introduce a character χ<sup><em>G</em></sup><sub style="position: relative; left: -.5em;">λ</sub> into the integrand. We show that the value of E<sub><em>G</em></sub>χ<sup><em>G</em></sup><sub style="position: relative; left: -.5em;">λ</sub>Φ<sub><em>n</em></sub>/E<sub><em>G</em></sub>Φ<sub><em>n</em></sub> approaches a constant for large <em>n</em>. More surprisingly, the ratio we obtain only changes with Φ<sub><em>n</em></sub> and λ and is independent of the Cartan type of <em>G</em>. Even in the unitary case, Bump and Diaconis [Daniel Bump, Persi Diaconis, Toeplitz minors, J. Combin. Theory Ser. A 97 (2) (2002) 252-271. Erratum for the proof of Theorem 4 available at http://sporadic.stanford.edu/bump/correction.ps and in a third reference in the abstract] have obtained the same ratio. Finally, those ratios can be combined with asymptotics for E<sub><em>G</em></sub>Φ<sub><em>n</em></sub> due to Johansson [Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (3) (1997) 519-545] and provide asymptotics for E<sub><em>G</em></sub>χ<sup><em>G</em></sup><sub style="position: relative; left: -.5em;">λ</sub>Φ<sub><em>n</em></sub>. © 2007 Elsevier Inc. All rights reserved. |
| first_indexed | 2024-03-06T18:59:39Z |
| format | Journal article |
| id | oxford-uuid:13126910-61bb-40f3-8f0c-61be00c85de2 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T18:59:39Z |
| publishDate | 2007 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | oxford-uuid:13126910-61bb-40f3-8f0c-61be00c85de22022-03-26T10:11:47ZAverages over classical Lie groups, twisted by charactersJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:13126910-61bb-40f3-8f0c-61be00c85de2EnglishSymplectic Elements at OxfordElsevier2007Dehaye, PWe compute E<sub><em>G</em></sub>(∏<sub><em>i</em></sub>tr(g<sup>λ<sub><em>i</em></sub></sup>)), where <em>g</em> ∈ <em>G</em> = Sp (2<em>n</em>) or SO (<em>m</em>) (<em>m</em> = 2<em>n</em>, 2<em>n</em> + 1) with Haar measure. This was first obtained by Diaconis and Shahshahani [Persi Diaconis, Mehrdad Shahshahani, On the eigenvalues of random matrices, J. Appl. Probab. 31A (1994) 49-62. Studies in applied probability], but our proof is more self-contained and gives a combinatorial description for the answer. We also consider how averages of general symmetric functions E<sub><em>G</em></sub>Φ<sub><em>n</em></sub> are affected when we introduce a character χ<sup><em>G</em></sup><sub style="position: relative; left: -.5em;">λ</sub> into the integrand. We show that the value of E<sub><em>G</em></sub>χ<sup><em>G</em></sup><sub style="position: relative; left: -.5em;">λ</sub>Φ<sub><em>n</em></sub>/E<sub><em>G</em></sub>Φ<sub><em>n</em></sub> approaches a constant for large <em>n</em>. More surprisingly, the ratio we obtain only changes with Φ<sub><em>n</em></sub> and λ and is independent of the Cartan type of <em>G</em>. Even in the unitary case, Bump and Diaconis [Daniel Bump, Persi Diaconis, Toeplitz minors, J. Combin. Theory Ser. A 97 (2) (2002) 252-271. Erratum for the proof of Theorem 4 available at http://sporadic.stanford.edu/bump/correction.ps and in a third reference in the abstract] have obtained the same ratio. Finally, those ratios can be combined with asymptotics for E<sub><em>G</em></sub>Φ<sub><em>n</em></sub> due to Johansson [Kurt Johansson, On random matrices from the compact classical groups, Ann. of Math. (2) 145 (3) (1997) 519-545] and provide asymptotics for E<sub><em>G</em></sub>χ<sup><em>G</em></sup><sub style="position: relative; left: -.5em;">λ</sub>Φ<sub><em>n</em></sub>. © 2007 Elsevier Inc. All rights reserved. |
| spellingShingle | Dehaye, P Averages over classical Lie groups, twisted by characters |
| title | Averages over classical Lie groups, twisted by characters |
| title_full | Averages over classical Lie groups, twisted by characters |
| title_fullStr | Averages over classical Lie groups, twisted by characters |
| title_full_unstemmed | Averages over classical Lie groups, twisted by characters |
| title_short | Averages over classical Lie groups, twisted by characters |
| title_sort | averages over classical lie groups twisted by characters |
| work_keys_str_mv | AT dehayep averagesoverclassicalliegroupstwistedbycharacters |