Superinduction for pattern groups
It is well known that the representation theory of the finite group of unipotent upper-triangular matrices U[subscript n] over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to th...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Elsevier
2015
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| Online Access: | http://hdl.handle.net/1721.1/96168 |
| _version_ | 1811076419565912064 |
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| author | Marberg, Eric Thiem, Nathaniel |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Marberg, Eric Thiem, Nathaniel |
| author_sort | Marberg, Eric |
| collection | MIT |
| description | It is well known that the representation theory of the finite group of unipotent upper-triangular matrices U[subscript n] over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies Diaconis–Isaacs' concept of superinduction in pattern groups. While superinduction shares many desirable properties with usual induction, it no longer takes characters to characters. We begin by finding sufficient conditions guaranteeing that superinduction is in fact induction. It turns out for two natural embeddings of U[subscript m] in U[subscript n], superinduction is induction. We conclude with an explicit combinatorial algorithm for computing this induction analogous to the Pieri-formulas for the symmetric group. |
| first_indexed | 2024-09-23T10:21:31Z |
| format | Article |
| id | mit-1721.1/96168 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T10:21:31Z |
| publishDate | 2015 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/961682022-09-26T17:23:34Z Superinduction for pattern groups Marberg, Eric Thiem, Nathaniel Massachusetts Institute of Technology. Department of Mathematics Marberg, Eric It is well known that the representation theory of the finite group of unipotent upper-triangular matrices U[subscript n] over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies Diaconis–Isaacs' concept of superinduction in pattern groups. While superinduction shares many desirable properties with usual induction, it no longer takes characters to characters. We begin by finding sufficient conditions guaranteeing that superinduction is in fact induction. It turns out for two natural embeddings of U[subscript m] in U[subscript n], superinduction is induction. We conclude with an explicit combinatorial algorithm for computing this induction analogous to the Pieri-formulas for the symmetric group. Stanford University 2015-03-25T14:50:15Z 2015-03-25T14:50:15Z 2009-04 2007-12 Article http://purl.org/eprint/type/JournalArticle 00218693 1090-266X http://hdl.handle.net/1721.1/96168 Marberg, Eric, and Nathaniel Thiem. “Superinduction for Pattern Groups.” Journal of Algebra 321, no. 12 (June 2009): 3681–3703. © 2009 Elsevier Inc. en_US http://dx.doi.org/10.1016/j.jalgebra.2009.03.003 Journal of Algebra Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Elsevier Elsevier |
| spellingShingle | Marberg, Eric Thiem, Nathaniel Superinduction for pattern groups |
| title | Superinduction for pattern groups |
| title_full | Superinduction for pattern groups |
| title_fullStr | Superinduction for pattern groups |
| title_full_unstemmed | Superinduction for pattern groups |
| title_short | Superinduction for pattern groups |
| title_sort | superinduction for pattern groups |
| url | http://hdl.handle.net/1721.1/96168 |
| work_keys_str_mv | AT marbergeric superinductionforpatterngroups AT thiemnathaniel superinductionforpatterngroups |