A Hopf-power Markov chain on compositions
In a recent paper, Diaconis, Ram and I constructed Markov chains using the coproduct-then-product map of a combinatorial Hopf algebra. We presented an algorithm for diagonalising a large class of these "Hopf-power chains", including the Gilbert-Shannon-Reeds model of riffle-shuffling of a...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2013-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/2316/pdf |
| _version_ | 1797270297585385472 |
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| author | C.Y. Amy Pang |
| author_facet | C.Y. Amy Pang |
| author_sort | C.Y. Amy Pang |
| collection | DOAJ |
| description | In a recent paper, Diaconis, Ram and I constructed Markov chains using the coproduct-then-product map of a combinatorial Hopf algebra. We presented an algorithm for diagonalising a large class of these "Hopf-power chains", including the Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards and a rock-breaking model. A very restrictive condition from that paper is removed in my thesis, and this extended abstract focuses on one application of the improved theory. Here, I use a new technique of lumping Hopf-power chains to show that the Hopf-power chain on the algebra of quasisymmetric functions is the induced chain on descent sets under riffle-shuffling. Moreover, I relate its right and left eigenfunctions to Garsia-Reutenauer idempotents and ribbon characters respectively, from which I recover an analogous result of Diaconis and Fulman (2012) concerning the number of descents under riffle-shuffling. |
| first_indexed | 2024-04-25T02:02:02Z |
| format | Article |
| id | doaj.art-a670e97e05524fd09cd4889e31e2e33f |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:02:02Z |
| publishDate | 2013-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-a670e97e05524fd09cd4889e31e2e33f2024-03-07T14:52:36ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502013-01-01DMTCS Proceedings vol. AS,...Proceedings10.46298/dmtcs.23162316A Hopf-power Markov chain on compositionsC.Y. Amy Pang0https://orcid.org/0000-0003-1191-4943Department of Mathematics [Stanford]In a recent paper, Diaconis, Ram and I constructed Markov chains using the coproduct-then-product map of a combinatorial Hopf algebra. We presented an algorithm for diagonalising a large class of these "Hopf-power chains", including the Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards and a rock-breaking model. A very restrictive condition from that paper is removed in my thesis, and this extended abstract focuses on one application of the improved theory. Here, I use a new technique of lumping Hopf-power chains to show that the Hopf-power chain on the algebra of quasisymmetric functions is the induced chain on descent sets under riffle-shuffling. Moreover, I relate its right and left eigenfunctions to Garsia-Reutenauer idempotents and ribbon characters respectively, from which I recover an analogous result of Diaconis and Fulman (2012) concerning the number of descents under riffle-shuffling.https://dmtcs.episciences.org/2316/pdfquasisymmetric functionsriffle shufflingdescent setcombinatorial hopf algebras[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | C.Y. Amy Pang A Hopf-power Markov chain on compositions Discrete Mathematics & Theoretical Computer Science quasisymmetric functions riffle shuffling descent set combinatorial hopf algebras [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | A Hopf-power Markov chain on compositions |
| title_full | A Hopf-power Markov chain on compositions |
| title_fullStr | A Hopf-power Markov chain on compositions |
| title_full_unstemmed | A Hopf-power Markov chain on compositions |
| title_short | A Hopf-power Markov chain on compositions |
| title_sort | hopf power markov chain on compositions |
| topic | quasisymmetric functions riffle shuffling descent set combinatorial hopf algebras [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2316/pdf |
| work_keys_str_mv | AT cyamypang ahopfpowermarkovchainoncompositions AT cyamypang hopfpowermarkovchainoncompositions |