 
Summary: COMMUTATION RELATIONS FOR ARBITRARY QUANTUM
MINORS
K. R. GOODEARL
Abstract. Complete sets of commutation relations for arbitrary pairs of quan
tum minors are computed, with explicit coefficients in closed form.
Introduction
The title of this paper begins with what may seem a misnomer the term com
mutation relation, in current usage, does not refer to a commutativity condition,
xy = yx, but has evolved to encompass various "skew commutativity" conditions
that have proved to be useful replacements for commutativity. Older types of com
mutation relations include conditions of the form xy  yx = z, used in defining
Weyl algebras and enveloping algebras. In quantized versions of classical algebras,
relations such as xy = qyx (known as qcommutation) appear, along with mix
tures of both types. Thus, it has become common to refer to any equation of the
form xy = yx + z, where is a nonzero scalar, as a commutation relation for
x and y. One important use of such a relation, especially in enveloping algebras,
is that if the algebra supports a filtration such that deg(z) < deg(x) + deg(y),
then the images of x and y in the associated graded algebra, call them x and y,
commute up to a scalar: xy = yx. Similarly, the cosets of x and y modulo the
ideal generated by z commute up to . Such coset relations are key ingredients in
