Fermionic Diagonal Coinvariants
| dc.contributor.advisor | Jim Haglund | |
| dc.contributor.advisor | Julia Hartmann | |
| dc.contributor.author | Kim, Jongwon | |
| dc.date | 2023-05-18T03:48:06.000 | |
| dc.date.accessioned | 2023-05-22T18:40:47Z | |
| dc.date.available | 2001-01-01T00:00:00Z | |
| dc.date.copyright | 2022-10-05T20:22:00-07:00 | |
| dc.date.issued | 2022-01-01 | |
| dc.date.submitted | 2022-10-05T08:54:03-07:00 | |
| dc.description.abstract | Let $W$ be a complex reflection group of rank $n$ acting on its reflection representation $V \cong \mb{C}^n$. The doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ induces an action on the quotient by the ideal generate by $W$-invariants with vanishing constant term $\FDR_W = \wedge (V \oplus V^*) / \langle \wedge (V \oplus V^*)^W_{+} \rangle$. We describe the bi-graded $W$-module structure of $\FDR_W$ and introduce a variant of Motzkin paths that descends to the standard monomial basis of $\FDR_W$ with respect to certain term order. The top degree of $\FDR_W$ exhibits the Narayana refinement of Catalan numbers. When $W = S_n$, the symmetric group, $\FDR_{S_n} \cong R_{n,0,2}$, where $R_{n,0,2}$ is the special case of the Boson-Fermionic diagonal coinvariants with two sets of Fermionic variables. In this case, the $(i,j)$-th degree component is a difference of Kronecker product of two hook Schur functions. In addition we consider a module $M_{n,m}$ spanned by $m$-ary strings of length $n$. When $m = 2$, as a vector space, $M_{n,2} \cong \mb{C}[X_n] / \langle x_1^2, \ldots, x_n^2 \rangle$. The trivial component of $\dr_n \otimes M_{n,2}$ is a weighted sum of $q,t$-Narayana numbers which is a different $q,t$-Catalan number than the alternant of $\dr_n$. At $t = 1$, the trivial component equals the inversion generating function for $321$-avoiding permutations. | |
| dc.description.degree | Doctor of Philosophy (PhD) | |
| dc.format.extent | 83 p. | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.uri | https://repository.upenn.edu/handle/20.500.14332/32254 | |
| dc.language | en | |
| dc.legacy.articleid | 7365 | |
| dc.legacy.fulltexturl | https://repository.upenn.edu/cgi/viewcontent.cgi?article=7365&context=edissertations&unstamped=1 | |
| dc.provenance | Received from ProQuest | |
| dc.rights | Jongwon Kim | |
| dc.source.issue | 5579 | |
| dc.source.journal | Publicly Accessible Penn Dissertations | |
| dc.source.status | published | |
| dc.subject.other | Mathematics | |
| dc.title | Fermionic Diagonal Coinvariants | |
| dc.type | Dissertation/Thesis | |
| digcom.date.embargo | 2001-01-01T00:00:00-08:00 | |
| digcom.identifier | edissertations/5579 | |
| digcom.identifier.contextkey | 31624826 | |
| digcom.identifier.submissionpath | edissertations/5579 | |
| digcom.type | dissertation | |
| dspace.entity.type | Publication | |
| upenn.graduate.group | Mathematics |
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