CHAPTER 0

Gaussian Binomial Coefficients

Many formulas in the theory of quantum groups (more precisely: of quantized

enveloping algebras) are analogues of formulas in the theory of enveloping alge-

bras. However, in making this transition one quite often has to replace the familiar

binomial coefficients by certain q-analogues that are called Gaussian binomial coef-

ficients. The purpose of this short chapter is to state their definition and to collect

a few of their properties.

0.1. Let v be an indeterminate over Q. We are going to work in the fraction

field Q(v) of the polynomial ring Q[v). However, it turns out that all elements

considered are actually contained in the subring

Z[t;,t;_1]

of Q(v).

Set for all a € Z

r

, va -v~a

a =

- 7 , - 1

We have obviously [0] = 0 and [a] ^ 0 for a ^ 0. Furthermore, [0] = 0 and

[—a] = —[a] for all a. If a 0, then

v

a~l

+y

a-3

+ t

+v-*+*+v-«+lm

This shows for all a (using [—a] = —[a]) that [a] € Z[v,v l]

Define the Gaussian binomial coefficients by

[a][a- l ] - - - [ a - n + l]

for all a, n £ Z with n 0, and by

[1][2]-..

n

(2)

= 1.

(3)

We have obviously [a] and

[—a] = —[a] one checks easily that

= ( - i ) n

1, and

—a-hn — 1

n

= 0 if 0 a n. Using

(4)

for all a and n. This shows in particular that

= ( - i )

n (5)

5

http://dx.doi.org/10.1090/gsm/006/02