{"title":"关于平面仿射群子群的拟单项式","authors":"N. Samaruk","doi":"10.30970/ms.59.1.3-11","DOIUrl":null,"url":null,"abstract":"Let $H$ be a subgroup of the plane affine group ${\\rm Aff}(2)$ considered with the natural action on the vector space of two-variable polynomials. The polynomial family $\\{ B_{m,n}(x,y) \\}$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $ \\{ x^m y^n \\} $ and $\\{ B_{m,n}(x,y) \\}$ have \\textit{identical} matrices. We obtain a criterion of quasi-monomiality for the case when the group $H$ is generated by rotations and translations in terms of exponential generating function for the polynomial family $\\{ B_{m,n}(x,y) \\}$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Quasi-monomials with respect to subgroups of the plane affine group\",\"authors\":\"N. Samaruk\",\"doi\":\"10.30970/ms.59.1.3-11\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $H$ be a subgroup of the plane affine group ${\\\\rm Aff}(2)$ considered with the natural action on the vector space of two-variable polynomials. The polynomial family $\\\\{ B_{m,n}(x,y) \\\\}$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $ \\\\{ x^m y^n \\\\} $ and $\\\\{ B_{m,n}(x,y) \\\\}$ have \\\\textit{identical} matrices. We obtain a criterion of quasi-monomiality for the case when the group $H$ is generated by rotations and translations in terms of exponential generating function for the polynomial family $\\\\{ B_{m,n}(x,y) \\\\}$.\",\"PeriodicalId\":37555,\"journal\":{\"name\":\"Matematychni Studii\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-03-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Matematychni Studii\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.30970/ms.59.1.3-11\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.59.1.3-11","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
Quasi-monomials with respect to subgroups of the plane affine group
Let $H$ be a subgroup of the plane affine group ${\rm Aff}(2)$ considered with the natural action on the vector space of two-variable polynomials. The polynomial family $\{ B_{m,n}(x,y) \}$ is called quasi-monomial with respect to $H$ if the group operators in two different bases $ \{ x^m y^n \} $ and $\{ B_{m,n}(x,y) \}$ have \textit{identical} matrices. We obtain a criterion of quasi-monomiality for the case when the group $H$ is generated by rotations and translations in terms of exponential generating function for the polynomial family $\{ B_{m,n}(x,y) \}$.