Plücker Relations on Schur Functions
Michael Kleber
DOI: 10.1023/A:1011201816304
Abstract
We present a set of algebraic relations among Schur functions which are a multi-time generalization of the 
discrete Hirota relations 
known to hold among the Schur functions of rectangular partitions. We prove the relations as an application of a technique for turning Plücker relations into statements about Schur functions and other objects with similar definitions as determinants. We also give a quantum analogue of the relations which incorporates spectral parameters. Our proofs are mostly algebraic, but the relations have a clear combinatorial side, which we discuss.
discrete Hirota relations known to hold among the Schur functions of rectangular partitions. We prove the relations as an application of a technique for turning Plücker relations into statements about Schur functions and other objects with similar definitions as determinants. We also give a quantum analogue of the relations which incorporates spectral parameters. Our proofs are mostly algebraic, but the relations have a clear combinatorial side, which we discuss.Pages: 199–211
Keywords: Schur function; plücker relation; Jacobi-trudi; quantum; Hirota relation
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References
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2. E. Frenkel and N.Yu. Reshetikhin, “The q-characters of representations of quantum affine algebras and deformations of W-algebras,” In “Recent developments in quantum affine algebras and related topics,” (Raleigh, NC, 1998), Contemp. Math. 248, 163-205. Amer. Math. Soc. Providence RI 1999.
3. A.N. Kirillov, “Completeness of states of the generalized Heisenberg magnet,” in Russian, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI ) 134 (1984), transl. in J. Soviet Math. 36 (1987), 115-128.
4. A.N. Kirillov and N.Yu. Reshetikhin, “Representations of Yangians and multiplicities of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras,” in Russian Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI ) 160 (1987), transl. in J. Soviet Math. 52 (1990), 3156-3164.
5. M. Kleber, “Polynomial relations among characters coming from quantum affine algebras,” Math. Research Lett. 5(6), (1998), 731-742.
6. A. Kuniba, T. Nakanashi, and J. Suzuki, “Functional relations in solvable lattice models: I. Functional relations and representation theory,” Internat. J. Modern Phys. A 9(30), (1994), 5215-5266.
7. A. Lascoux and P. Pragacz, “Ribbon Schur functions,” European J. Combin. 9(6), (1988), 561-574. PL \ddot UCKER RELATIONS ON SCHUR FUNCTIONS 211
8. B. Leclerc, “Powers of staircase Schur functions and symmetric analogues of Bessel polynomials,” Discrete Math. 153(1-3), (1996), 213-227.
9. O. Lipan, P. Wiegmann, and A. Zabrodin, “Fusion rules for quantum transfer matrices as a dynamical system on Grassman manifolds,” Modern Phys. Lett. A 12(19), (1997), 1369-1378.
10. T. Nakashima, “Crystal base and a generalization of the Littlewood-Richardson rule for the classical Lie algebras,” Comm. Math. Phys. 154 (1993), 215-243.