Bitableau bases for Garsia–Haiman modules of hollow type

Garsia–Haiman modules C [ X n , Y n ] / I γ are quotient rings in the variables X n = { x 1 , x 2 , … , x n } and Y n = { y 1 , y 2 , … , y n } that generalize the quotient ring C [ X n ] / I , where I is the ideal generated by the elementary symmetric polynomials e j ( X n ) for 1 ⩽ j ⩽ n . A bitableau basis for the Garsia–Haiman modules of hollow type is constructed. Applications of this basis to representation theory and other related polynomial spaces are considered.