Geodesic
Singular vectors of the~Ding--Iohara--Miki algebra
Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 1, pp. 3-32

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We review properties of generalized Macdonald functions arising from the AGT correspondence. In particular, we explain a coincidence between generalized Macdonald functions and singular vectors of a certain algebra $\mathcal{A}(N)$ obtained using the level-$(N,0)$ representation (horizontal representation) of the Ding–Iohara–Miki algebra. Moreover, we give a factored formula for the Kac determinant of $\mathcal{A}(N)$, which proves the conjecture that the Poincaré–Birkhoff–Witt-type vectors of the algebra $\mathcal{A}(N)$ form a basis in its representation space.
Keywords: AGT correspondence, Macdonald symmetric function, Ding–Iohara–Miki algebra, singular vector. 
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     author = {Y. Ohkubo},
     title = {Singular vectors of {the~Ding--Iohara--Miki} algebra},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     number = {1},
     year = {2019},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2019_199_1_a0/}
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Y. Ohkubo. Singular vectors of the~Ding--Iohara--Miki algebra. Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 1, pp. 3-32. http://geodesic.mathdoc.fr/item/TMF_2019_199_1_a0/