Proceedings of the 44th Symposium on Ring Theory and ...

e(n, s) n = 6 7 8
s = 1 36 63 120
2 108 315 945
3 72 336 1575
4 0 63 675
e 0 (n, s) n = 6 7 8
s = 1 7 16 44
2 35 120 462
3 35 170 924
4 0 40 462
Table 0.1. The values <strong>of</strong> e(n, s) and e 0 (n, s)
but essentially counting <strong>the</strong> number <strong>of</strong> regular PVs associated with. Our main <strong>the</strong>orem
is <strong>the</strong> following:
Theorem 0.1. Let Q be a quiver <strong>of</strong> type A n with n ≥ 1 (resp. D n with n ≥ 4, E n with
n = 6, 7, 8). Then <strong>the</strong> number a(n, s) (resp. d(n, s), e(n, s)) <strong>of</strong> <strong>the</strong> isomorphic classes <strong>of</strong>
basic hom-orthogonal tilting KQ-modules having s pairwise non-isomorphic indecomposable
direct summands is given explicitely by <strong>the</strong> following:
(0.1)
(0.2)
(n + 1)!
a(n, s) =
s! (s + 1)! (n + 1 − 2s)!
= C s · ( n+1
2s )
if 1 ≤ s ≤ (n + 1)/2, and a(n, s) = 0 if o<strong>the</strong>rwise. Here C s = ( 2s
s ) /(s + 1) denotes <strong>the</strong>
s-th Catalan number.
d(n, s) =
(n − 1)!
(s!) 2 (n + 2 − 2s)! · {s 2 (s − 1) + n(n + 1 − 2s)(n + 2 − 2s) }
if 1 ≤ s ≤ (n + 2)/2, and d(n, s) = 0 if o<strong>the</strong>rwise. The values <strong>of</strong> e(n, s) for 1 ≤ s ≤
(n + 1)/2 are given in Table 0.1, and we have e(n, s) = 0 if o<strong>the</strong>rwise.
Our approach to this <strong>the</strong>orem, which was inspired by Seidel’s paper [7], is based on
an observation <strong>of</strong> perpendicular categories introduced by Sch<strong>of</strong>ield [6]. Here we point
out that <strong>the</strong> totality <strong>of</strong> a(n, s) or d(n, s) for fixed n can be expressed as a special value
<strong>of</strong> a hypergeometric function. As mentioned in Remark 2.4, <strong>the</strong> formula (0.2) has a
combinatorial interpretation.
According to Happel [4], if a Λ-module corresponding to a point contained in <strong>the</strong> dense
orbit <strong>of</strong> a PV (GL(d), Rep(Q, d)) has s pairwise non-isomorphic indecomposable direct
summands, <strong>the</strong>n <strong>the</strong> PV has exactly n − s basic relative invariants. Thus we obtain a
consequence <strong>of</strong> Theorem 0.1.
Corollary 0.2. Each regular PV associated with a quiver <strong>of</strong> type A n (resp. D n , E 6 , E 7 ,
and E 8 ) has at least (n − 1)/2 (resp. (n − 2)/2, 3, 3, and 4) basic relative invariants.
We say that X ∈ mod Λ is sincere if its dimension vector dim X does not have zero entry.
Sincere modules are fairly interesting to <strong>the</strong> <strong>the</strong>ory <strong>of</strong> PVs, because (GL(d), Rep(Q, d))
with non-sincere dimension can be regarded as a direct sum <strong>of</strong> at least two PVs associated
with proper subgraphs <strong>of</strong> Q. So we have counted <strong>the</strong>m:
Theorem 0.3. Let Q be a quiver <strong>of</strong> type A n with n ≥ 1 (resp. D n with n ≥ 4, E n with
n = 6, 7, 8). Then <strong>the</strong> number a 0 (n, s) (resp. d 0 (n, s), e 0 (n, s)) <strong>of</strong> <strong>the</strong> isomorphic classes
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