Quantum finite W-algebras for gl_N

A major contribution to the theory of quantum finite W-algebras in type A comes from the work of J. Brundan and A. Kleshchev who, investigating the relationship between W-algebras and Yangians, achieved important results concerning both their structure and their representation theory. In this framework, for a quantum finite W-algebra in type A, associated to any nilpotent element and arbitrary good grading, A. De Sole, V. Kac and D. Valeri constructed a matrix of Yangian type L(z) which encodes its generators and relations, generalizing the results of the same authors for classical affine W-algebras. We can then express L(z) in a nicer form: when the good grading is associated to a pyramid that is aligned to the right or to the left, we use a recursive formula to explicitly construct a matrix W(z) which provides us with a finite set of generators for the W-algebra satisfying Premet's conditions, and prove that the matrix L(z) can be obtained as a generalized quasideterminant of W(z). Finally, we explain how to generalize these results to an arbitrary good grading (and an arbitrary choice of an isotropic subspace), using fundamental results about the structure of quantum finite W-algebras due to W.L. Gan and V. Ginzburg, and J. Brundan and S. Goodwin.