6 I. PRELIMINARIES ON NON-CROSSING PARTITIONS
Then Si is separated by S2 and hence (7r5;)c = 7Ti for all TTI e NC(Si), e.g., for
7n = {(2,13),(4,6,10)}
we have
*? = {(1,14), (3,11,12), (5), (7,8,9)} and (*rf)c - {(2,13), (4,6,10)} =
Vl
.
On the other hand, 52 is not separated by Si, namely the pairs (7,8), (7,9), (8,9),
(11,12), and (1,14) cannot be separated by pairs from Si. So, e.g., for
TT2 = {(1), (3,11,14), (5,8), (7), (9), (12)}
we have
7T| = {(2), (4,10), (6), (13)} and ( ^ ) c = {(1,3,11,12,14), (5,7,8,9)} TT2.
1.1.6. REMARKS. 1) The lattice of non-crossing partitions was introduced by
Kreweras [Kre] in 1972. Since then there have appeared some purely combinatorial
investigations on this lattice, e.g. [Pou,Edel,Ede2,BSS,ES,Sim,SU,Bia3], but the
connection of this lattice with free products and free probability theory, as we
will use it here, is quite new and was realized for the first time in [Spe4]. This
connection between non-crossing partitions and free probability theory served also
as one starting point for Nica's investigations of g-deformations of free convolution
[Nic2,Nic3].
2) In [Spel,Spe2], we used the notion 'admissible' instead of 'non-crossing'.
3) In the case of an alternating decomposition, the map TTI K- 7rf was introduced
by Kreweras [Kre], see also the proof of Simion and Ullman [SU] that the lattice
NC(n) is self-dual.
1.2. Incidence algebra and convolution
Now we will use the lattice structure for defining a convolution for functions on
the lattice of non-crossing partitions (not to be mixed up with the later definition of
free convolution). This is a standard procedure for such lattices or, more generally,
for partially ordered sets (posets), see [Rot 1,Rot2].
1.2.1. DEFINITION. 1) The (large) incidence algebra I2 is defined as the set of
all complex-valued functions 77(71-, a) on {Jnew(NC(n) x NC(n)) with the property
that 77(71-, a) = 0 if IT ^ a. The set I2 becomes an algebra under the convolution
: I
2
x I
2
- I2
(0,77) i- e 77,
where
(0 * 77)(TT, a) := ] T 9(TT, v)r){y, a)
veNC{n)
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