{"title":"sl 2 ${mathfrak {sl}_2}$ qKZ 方程模为整数的解","authors":"Evgeny Mukhin, Alexander Varchenko","doi":"10.1112/jlms.12884","DOIUrl":null,"url":null,"abstract":"<p>We study the <i>qKZ</i> difference equations with values in the <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>th tensor power of the vector <span></span><math>\n <semantics>\n <msub>\n <mi>sl</mi>\n <mn>2</mn>\n </msub>\n <annotation>${\\mathfrak {sl}_2}$</annotation>\n </semantics></math> representation <span></span><math>\n <semantics>\n <mi>V</mi>\n <annotation>$V$</annotation>\n </semantics></math>, variables <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>z</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mi>⋯</mi>\n <mo>,</mo>\n <msub>\n <mi>z</mi>\n <mi>n</mi>\n </msub>\n </mrow>\n <annotation>$z_1,\\dots,z_n$</annotation>\n </semantics></math>, and integer step <span></span><math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math>. For any integer <span></span><math>\n <semantics>\n <mi>N</mi>\n <annotation>$N$</annotation>\n </semantics></math> relatively prime to the step <span></span><math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math>, we construct a family of polynomials <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>f</mi>\n <mi>r</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>z</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$f_r(z)$</annotation>\n </semantics></math> in variables <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>z</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mi>⋯</mi>\n <mo>,</mo>\n <msub>\n <mi>z</mi>\n <mi>n</mi>\n </msub>\n </mrow>\n <annotation>$z_1,\\dots,z_n$</annotation>\n </semantics></math> with values in <span></span><math>\n <semantics>\n <msup>\n <mi>V</mi>\n <mrow>\n <mo>⊗</mo>\n <mi>n</mi>\n </mrow>\n </msup>\n <annotation>$V^{\\otimes n}$</annotation>\n </semantics></math> such that the coordinates of these polynomials with respect to the standard basis of <span></span><math>\n <semantics>\n <msup>\n <mi>V</mi>\n <mrow>\n <mo>⊗</mo>\n <mi>n</mi>\n </mrow>\n </msup>\n <annotation>$V^{\\otimes n}$</annotation>\n </semantics></math> are polynomials with integer coefficients. We show that <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>f</mi>\n <mi>r</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>z</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$f_r(z)$</annotation>\n </semantics></math> satisfy the <i>qKZ</i> equations modulo <span></span><math>\n <semantics>\n <mi>N</mi>\n <annotation>$N$</annotation>\n </semantics></math>. Polynomials <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>f</mi>\n <mi>r</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>z</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$f_r(z)$</annotation>\n </semantics></math> are modulo <span></span><math>\n <semantics>\n <mi>N</mi>\n <annotation>$N$</annotation>\n </semantics></math> analogs of the hypergeometric solutions of the <i>qKZ</i> given in the form of multidimensional Barnes integrals.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Solutions of the \\n \\n \\n sl\\n 2\\n \\n ${\\\\mathfrak {sl}_2}$\\n qKZ equations modulo an integer\",\"authors\":\"Evgeny Mukhin, Alexander Varchenko\",\"doi\":\"10.1112/jlms.12884\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study the <i>qKZ</i> difference equations with values in the <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math>th tensor power of the vector <span></span><math>\\n <semantics>\\n <msub>\\n <mi>sl</mi>\\n <mn>2</mn>\\n </msub>\\n <annotation>${\\\\mathfrak {sl}_2}$</annotation>\\n </semantics></math> representation <span></span><math>\\n <semantics>\\n <mi>V</mi>\\n <annotation>$V$</annotation>\\n </semantics></math>, variables <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>z</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <mi>⋯</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>z</mi>\\n <mi>n</mi>\\n </msub>\\n </mrow>\\n <annotation>$z_1,\\\\dots,z_n$</annotation>\\n </semantics></math>, and integer step <span></span><math>\\n <semantics>\\n <mi>κ</mi>\\n <annotation>$\\\\kappa$</annotation>\\n </semantics></math>. For any integer <span></span><math>\\n <semantics>\\n <mi>N</mi>\\n <annotation>$N$</annotation>\\n </semantics></math> relatively prime to the step <span></span><math>\\n <semantics>\\n <mi>κ</mi>\\n <annotation>$\\\\kappa$</annotation>\\n </semantics></math>, we construct a family of polynomials <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>f</mi>\\n <mi>r</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>z</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$f_r(z)$</annotation>\\n </semantics></math> in variables <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>z</mi>\\n <mn>1</mn>\\n </msub>\\n <mo>,</mo>\\n <mi>⋯</mi>\\n <mo>,</mo>\\n <msub>\\n <mi>z</mi>\\n <mi>n</mi>\\n </msub>\\n </mrow>\\n <annotation>$z_1,\\\\dots,z_n$</annotation>\\n </semantics></math> with values in <span></span><math>\\n <semantics>\\n <msup>\\n <mi>V</mi>\\n <mrow>\\n <mo>⊗</mo>\\n <mi>n</mi>\\n </mrow>\\n </msup>\\n <annotation>$V^{\\\\otimes n}$</annotation>\\n </semantics></math> such that the coordinates of these polynomials with respect to the standard basis of <span></span><math>\\n <semantics>\\n <msup>\\n <mi>V</mi>\\n <mrow>\\n <mo>⊗</mo>\\n <mi>n</mi>\\n </mrow>\\n </msup>\\n <annotation>$V^{\\\\otimes n}$</annotation>\\n </semantics></math> are polynomials with integer coefficients. We show that <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>f</mi>\\n <mi>r</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>z</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$f_r(z)$</annotation>\\n </semantics></math> satisfy the <i>qKZ</i> equations modulo <span></span><math>\\n <semantics>\\n <mi>N</mi>\\n <annotation>$N$</annotation>\\n </semantics></math>. Polynomials <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>f</mi>\\n <mi>r</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>z</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$f_r(z)$</annotation>\\n </semantics></math> are modulo <span></span><math>\\n <semantics>\\n <mi>N</mi>\\n <annotation>$N$</annotation>\\n </semantics></math> analogs of the hypergeometric solutions of the <i>qKZ</i> given in the form of multidimensional Barnes integrals.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-03-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12884\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12884","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们研究了在向量 sl 2 ${mathfrak {sl}_2}$ 表示 V $V$ 的 n $n$ 张量幂中取值的 qKZ 差分方程,变量 z 1 , ⋯ , z n $z_1,\dots,z_n$ 以及整数步长 κ $\kappa$ 。对于与步长 κ $kappa$ 相对质数的任意整数 N $N$ ,我们构造了变量 z 1 , ⋯ , z n $z_1,\dots,z_n$ 在 V ⊗ n $V^{otimes n}$ 中取值的多项式 f r ( z ) $f_r(z)$ 族,使得这些多项式相对于 V ⊗ n $V^{otimes n}$ 的标准基的坐标是具有整数系数的多项式。我们证明 f r ( z ) $f_r(z)$ 满足模为 N $N$ 的 qKZ 方程。多项式 f r ( z ) $f_r(z)$ 是以多维巴恩斯积分形式给出的 qKZ 超几何解的 N $N$ 模类似物。
Solutions of the
sl
2
${\mathfrak {sl}_2}$
qKZ equations modulo an integer
We study the qKZ difference equations with values in the th tensor power of the vector representation , variables , and integer step . For any integer relatively prime to the step , we construct a family of polynomials in variables with values in such that the coordinates of these polynomials with respect to the standard basis of are polynomials with integer coefficients. We show that satisfy the qKZ equations modulo . Polynomials are modulo analogs of the hypergeometric solutions of the qKZ given in the form of multidimensional Barnes integrals.
期刊介绍:
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