純粋・応用数学研究センター

組合せ論セミナー

直井 克之（東京農工大学）「Jacobi-Trudi formula and quantum loop algebras」

It is known that Schur polynomials (distinguished basis of symmetric polynomials) can be expressed as a determinant of a matrix whose components are complete symmetric polynomials, and this formula is called the Jacobi-Trudi formula. Since a Schur polynomial coincides with the character of a simple GL_n-module, we can also regard the formula as a character formula on simple GL_n-modules. A similar character formula does not hold for simple modules over other Lie groups such as SO_{2n} or Sp_{2n}. However, we can show that a similar formula does hold for a simple module over a quantum loop algebra. In other words, we can generalize the Jacobi-Trudi formula to other types using a quantum loop algebra. In this talk we will briefly review the Jacobi-Trudi formula first, and then explain how to generalize the formula to other types by considering simple modules over a quantum loop algebra.