All Kronecker coefficients are reduced Kronecker coefficients
We settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. T...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Pi |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508624000234/type/journal_article |
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| author | Christian Ikenmeyer Greta Panova |
| author_facet | Christian Ikenmeyer Greta Panova |
| author_sort | Christian Ikenmeyer |
| collection | DOAJ |
| description | We settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of an open problem by Stanley from 2000 and an open problem by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is
${\textsf {NP}}$
-hard, and computing them is
${{{\textsf {#P}}}}$
-hard under parsimonious many-one reductions. Our proof also provides an explicit isomorphism of the corresponding highest weight vector spaces. |
| format | Article |
| id | doaj-art-b96ff9b27b59461eb639e3e0fe9b74b1 |
| institution | Kabale University |
| issn | 2050-5086 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj-art-b96ff9b27b59461eb639e3e0fe9b74b12024-11-18T10:15:25ZengCambridge University PressForum of Mathematics, Pi2050-50862024-01-011210.1017/fmp.2024.23All Kronecker coefficients are reduced Kronecker coefficientsChristian Ikenmeyer0https://orcid.org/0000-0003-4654-177XGreta Panova1https://orcid.org/0000-0003-0785-1580Department of Computer Science, Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom; E-mail:Mathematics Department, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089, USAWe settle the question of where exactly do the reduced Kronecker coefficients lie on the spectrum between the Littlewood-Richardson and Kronecker coefficients by showing that every Kronecker coefficient of the symmetric group is equal to a reduced Kronecker coefficient by an explicit construction. This implies the equivalence of an open problem by Stanley from 2000 and an open problem by Kirillov from 2004 about combinatorial interpretations of these two families of coefficients. Moreover, as a corollary, we deduce that deciding the positivity of reduced Kronecker coefficients is ${\textsf {NP}}$ -hard, and computing them is ${{{\textsf {#P}}}}$ -hard under parsimonious many-one reductions. Our proof also provides an explicit isomorphism of the corresponding highest weight vector spaces.https://www.cambridge.org/core/product/identifier/S2050508624000234/type/journal_article20C3020C1520G0522E4605E05 |
| spellingShingle | Christian Ikenmeyer Greta Panova All Kronecker coefficients are reduced Kronecker coefficients Forum of Mathematics, Pi 20C30 20C15 20G05 22E46 05E05 |
| title | All Kronecker coefficients are reduced Kronecker coefficients |
| title_full | All Kronecker coefficients are reduced Kronecker coefficients |
| title_fullStr | All Kronecker coefficients are reduced Kronecker coefficients |
| title_full_unstemmed | All Kronecker coefficients are reduced Kronecker coefficients |
| title_short | All Kronecker coefficients are reduced Kronecker coefficients |
| title_sort | all kronecker coefficients are reduced kronecker coefficients |
| topic | 20C30 20C15 20G05 22E46 05E05 |
| url | https://www.cambridge.org/core/product/identifier/S2050508624000234/type/journal_article |
| work_keys_str_mv | AT christianikenmeyer allkroneckercoefficientsarereducedkroneckercoefficients AT gretapanova allkroneckercoefficientsarereducedkroneckercoefficients |