{"title":"外代数的拟对称调和","authors":"N. Bergeron, K. Chan, F. Soltani, M. Zabrocki","doi":"10.4153/S0008439523000024","DOIUrl":null,"url":null,"abstract":"Abstract We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let \n$R_n$\n denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in \n$R_n$\n form a commutative subalgebra of \n$R_n$\n . (2) There is a basis of the quotient of \n$R_n$\n by the ideal \n$I_n$\n generated by the quasisymmetric polynomials in \n$R_n$\n that is indexed by ballot sequences. The Hilbert series of the quotient is given by \n$$ \\begin{align*}\\text{Hilb}_{R_n/I_n}(q) = \\sum_{k=0}^{\\lfloor{n/2}\\rfloor} f^{(n-k,k)} q^k\\,,\\end{align*} $$\n where \n$f^{(n-k,k)}$\n is the number of standard tableaux of shape \n$(n-k,k)$\n . (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Quasisymmetric harmonics of the exterior algebra\",\"authors\":\"N. Bergeron, K. Chan, F. Soltani, M. Zabrocki\",\"doi\":\"10.4153/S0008439523000024\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let \\n$R_n$\\n denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in \\n$R_n$\\n form a commutative subalgebra of \\n$R_n$\\n . (2) There is a basis of the quotient of \\n$R_n$\\n by the ideal \\n$I_n$\\n generated by the quasisymmetric polynomials in \\n$R_n$\\n that is indexed by ballot sequences. The Hilbert series of the quotient is given by \\n$$ \\\\begin{align*}\\\\text{Hilb}_{R_n/I_n}(q) = \\\\sum_{k=0}^{\\\\lfloor{n/2}\\\\rfloor} f^{(n-k,k)} q^k\\\\,,\\\\end{align*} $$\\n where \\n$f^{(n-k,k)}$\\n is the number of standard tableaux of shape \\n$(n-k,k)$\\n . (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-06-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4153/S0008439523000024\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4153/S0008439523000024","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Abstract We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let
$R_n$
denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in
$R_n$
form a commutative subalgebra of
$R_n$
. (2) There is a basis of the quotient of
$R_n$
by the ideal
$I_n$
generated by the quasisymmetric polynomials in
$R_n$
that is indexed by ballot sequences. The Hilbert series of the quotient is given by
$$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$
where
$f^{(n-k,k)}$
is the number of standard tableaux of shape
$(n-k,k)$
. (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.