{"id":{"repo_id":"penn","oai_identifier":"oai:repository.upenn.edu:20.500.14332/32254"},"canonical_url":"https://search.dev.ndltd.org/etd/penn/oai:repository.upenn.edu:20.500.14332/32254","repository":{"repo_id":"penn","name":"University of Pennsylvania","base_url":"https://repository.upenn.edu/server/oai/request"},"display":{"title":"Fermionic Diagonal Coinvariants","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.","abstract_html":"Let $W$ be a complex reflection group of rank $n$ acting on its reflection representation <span class=\"etd-inline-math\">V \\cong \\mb{C}<sup>n</sup></span>. The doubly graded action of $W$ on the exterior algebra <span class=\"etd-inline-math\">\\wedge (V \\oplus V<sup>*</sup>)</span> induces an action on the quotient by the ideal generate by $W$-invariants with vanishing constant term <span class=\"etd-inline-math\">\\FDR<sub>W</sub> = \\wedge (V \\oplus V<sup>*</sup>) / \\langle \\wedge (V \\oplus V<sup>*</sup>)<sup>W</sup><sub>+</sub> \\rangle</span>. We describe the bi-graded $W$-module structure of <span class=\"etd-inline-math\">\\FDR<sub>W</sub></span> and introduce a variant of Motzkin paths that descends to the standard monomial basis of <span class=\"etd-inline-math\">\\FDR<sub>W</sub></span> with respect to certain term order. The top degree of <span class=\"etd-inline-math\">\\FDR<sub>W</sub></span> exhibits the Narayana refinement of Catalan numbers. When <span class=\"etd-inline-math\">W = S<sub>n</sub></span>, the symmetric group, <span class=\"etd-inline-math\">\\FDR<sub>S<sub>n</sub></sub> \\cong R<sub>n,0,2</sub></span>, where <span class=\"etd-inline-math\">R<sub>n,0,2</sub></span> 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 <span class=\"etd-inline-math\">M<sub>n,m</sub></span> spanned by $m$-ary strings of length $n$. When $m = 2$, as a vector space, <span class=\"etd-inline-math\">M<sub>n,2</sub> \\cong \\mb{C}[X<sub>n</sub>] / \\langle x<sub>1</sub><sup>2</sup>, \\ldots, x<sub>n</sub><sup>2</sup> \\rangle</span>. The trivial component of <span class=\"etd-inline-math\">\\dr<sub>n</sub> \\otimes M<sub>n,2</sub></span> is a weighted sum of $q,t$-Narayana numbers which is a different $q,t$-Catalan number than the alternant of <span class=\"etd-inline-math\">\\dr<sub>n</sub></span>. At $t = 1$, the trivial component equals the inversion generating function for $321$-avoiding permutations.","abstract_has_math":true,"creators":["Kim, Jongwon"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Jim Haglund","Julia Hartmann"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022","date_published":"2022","updated_at":"2026-07-24T03:46:11Z","subjects":[],"languages":["en"],"rights":["Jongwon Kim"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://repository.upenn.edu/handle/20.500.14332/32254","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Jim Haglund","Julia Hartmann"]},{"key":"dc:creator","label":"Author","values":["Kim, Jongwon"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-05-18T03:48:06.000"]},{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2023-05-22T18:40:47Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2001-01-01T00:00:00Z"]},{"key":"dc:date.issued","label":"Date","values":["2022"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation/Thesis"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Jongwon Kim"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://repository.upenn.edu/handle/20.500.14332/32254"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Doctor of Philosophy (PhD)"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Fermionic Diagonal Coinvariants"]}]}],"canonical_facts":{"dc:contributor.advisor":["Jim Haglund","Julia Hartmann"],"dc:creator":["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.issued":["2022"],"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.mimetype":["application/pdf"],"dc:identifier.uri":["https://repository.upenn.edu/handle/20.500.14332/32254"],"dc:language":["en"],"dc:rights":["Jongwon Kim"],"dc:title":["Fermionic Diagonal Coinvariants"],"dc:type":["Dissertation/Thesis"]},"updated_at":"2026-07-24T03:46:11Z"}