Supersymmetric Polynomials and the Center of the Walled Brauer Algebra
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Supersymmetric Polynomials and the Center of the Walled Brauer Algebra Ji Hye Jung1 · Myungho Kim2 Received: 14 November 2018 / Accepted: 15 August 2019 / © Springer Nature B.V. 2019
Abstract We study a commuting family of elements of the walled Brauer algebra Br,s (δ), called the Jucys-Murphy elements, and show that the supersymmetric polynomials in these elements belong to the center of the walled Brauer algebra. When Br,s (δ) is semisimple, we show that those supersymmetric polynomials generate the center. In addition, we present an analogue of Jucys-Murphy elements for the quantized walled Brauer algebra Hr,s (q, ρ) over C(q, ρ) and by taking the classical limit we show that the supersymmetric polynomials in these elements generates the center. It follows that H. Morton’s conjecture, which appeared in the study of the relation between the framed HOMFLY skein on the annulus and that on the rectangle with designated boundary points, holds if we extend the scalar from Z[q ±1 , ρ ±1 ](q−q −1 ) to C(q, ρ). Keywords Walled Brauer algebra · Center · Jucys-Murphy element · Supersymmetric polynomials · Quantized walled Brauer algebra Mathematics Subject Classification (2010) 16U70 · 16T30 · 05E05
Presented by: Anne Schilling This research was supported by Sogang Research Team for Discrete and Geometric Structures and by NRF-2013R1A1A2063671. This research was supported by the National Research Foundation of Korea(NF) grant funded by the Korea government(MSIP) (No. NRF-2017R1C1B2007824). The authors were supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine Myungho Kim
[email protected] Ji Hye Jung [email protected] 1
Department of Mathematics, Sogang University, Seoul, 04107, Korea
2
Department of Mathematics, Kyung Hee University, Seoul, 02447, Korea
J.H. Jung, M. Kim
1 Introduction The Jucys-Murphy elements of the group algebra C[Sr ] of the symmetric group of r letters are given by Lk :=
k−1
(j, k) (1 ≤ k ≤ r),
j =1
where (a, b) denotes the transposition exchanging a and b for 1 ≤ a, b ≤ r. In particular, L1 = 0 and Lk ’s are commuting to each other. These elements were introduced independently in [12, 25] and it was shown that the center of C[Sr ] consists of all the symmetric polynomials in these elements [12, 26]. This remarkable fact leads various interesting studies. For example, the ring homomorphism from the ring of symmetric polynomials to the center of C[Sr ], which is called the Jucys-Murphy specialization, has been studied by many researchers. Let f (x1 , . . . , xr ) be a symmetric polynomial. Since the evaluation f (L1 , · · · , Lr ) belongs to the center of C[Sr ], it canbe written uniquely as a linear combination of the natural basis Cμ | μ is a partition of r of the center, where Cμ denotes the sum of all permutations with the same cycle type μ. That is, in the center of C[Sr ], we f f have an equation f (L1 , . . . , Lr ) = μ aμ Cμ (aμ ∈ C). The problem to calculate coeff
ficients aμ for
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