We construct the shifted genus expanded algebra, which is isomorphic to the central subalgebra of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra Λ* defined by Okounkov and Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers; thus, we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.
REFERENCES
1.
Alexandrov
, A.
, Mironov
, A.
, Morozov
, A.
, and Natanzon
, S.
, “On KP-integrable Hurwitz functions
,” J. High Energy Phys.
2014
(11
), 080
.2.
Goulden
, I. P.
and Jackson
, D. M.
, “A proof of a conjecture for the number of ramified covering of the sphere by the torus
,” J. Comb. Theory, Ser. A
88
, 246
–258
(1999
).3.
Goulden
, I. P.
and Jackson
, D. M.
, “The number of ramified covering of the sphere by the double torus, and a general form for higher genera
,” J. Comb. Theory, Ser. A
88
, 259
–275
(1999
).4.
Goulden
, I. P.
and Jackson
, D. M.
, “Transitive factorisations into transpositions and holomorphic mapping on the sphere
,” Proc. Am. Math. Soc.
125
, 51
–60
(1997
).5.
Hurwitz
, A.
, “Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten
,” Math. Ann.
39
, 1
–60
(1891
).6.
Ivanov
, V. N.
and Kerov
, S. V. E.
, “The algebra of conjugacy classes in symmetric groups, and partial permutations
,” J. Math. Sci. (New York)
107
(5
), 4212
–4230
(2001
).7.
Ionel
, E.
and Parker
, T.
, “Gromov-Witten invariants of symplectic sums
,” Math. Res. Lett.
5
, 563
–576
(1998
).8.
Lerman
, E.
, “Symplectic cuts
,” Math. Res. Lett.
2
, 247
–258
(1995
).9.
Li
, A. M.
and Ruan
, Y. B.
, “Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds
,” Invent. Math.
145
, 151
–218
(2001
).10.
Li
, A. M.
, Zhao
, G. S.
, and Zheng
, Q.
, “The number of ramified covering of a Riemann surface by Riemann surface
,” Commun. Math. Phys.
213
, 685
–696
(2000
).11.
Macdonald
, I. G.
, Symmetric Functions and Hall Polynomials
, 2nd ed. Oxford Mathematical Monographs, Oxford Science Publications
(The Clarendon Press, Oxford University Press
, New York
, 1995
), With contributions by A. Zelevinsky.12.
Morozov
, A.
and Natanzon
, S.
, “Complete set of cut-and-join operators in Hurwitz-Kontsevich theory
,” Theoret. Math. Phys.
166
(1
), 1
–22
(2011
).13.
Mironov
, A.
, Morozov
, A.
, and Natanzon
, S.
, “Universal algebras of Hurwitz numbers,” e-print arXiv:0909.1164.14.
Mironov
, A.
, Morozov
, A.
, and Natanzon
, S.
, “Algebra of differential operators associated with Young diagrams
,” J. Geom. Phys.
62
, 148
–155
(2012
).15.
Morozov
, A.
and Shakirov
, Sh.
, “Generation of Matrix Models by -operators
,” J. High Energy Phys.
4
(4
), 33
(2009
).16.
Okounkov
, A. Y.
and Olshanskii
, G. I.
, “Shifted Schur functions
,” Algebra Anal.
9
(2
), 73
–146
(1997
).17.
Okounkov
, A.
and Pandharipande
, R.
, “Gromov-Witten theory, Hurwitz theory, and completed cycles
,” Ann. Math.
163
, 517
–560
(2006
).18.
Zheng
, Q.
, “Genus expanded cut-and-join operators and generalized Hurwtiz numbers
,” Acta Math. Sin.
(unpublished); e-print arXiv:1511.00823.© 2016 Author(s).
2016
Author(s)
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