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
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