Table Of ContentDEMYSTIFYING A DIVISIBILITY PROPERTY OF THE
KOSTANT PARTITION FUNCTION
1 KAROLAME´SZA´ROS
1
0
2 Abstract. Westudyafamilyofidentitiesregardingadivisibilityprop-
n ertyoftheKostantpartitionfunctionwhichfirstappearedinapaperof
a Baldoni and Vergne. To prove the identities, Baldoni and Vergne used
J techniquesofresiduesandcalledtheresultingdivisibilityproperty“mys-
2 terious.” Weprovetheseidentitiesentirely combinatorially andprovide
a natural explanation of why the divisibility occurs. We also point out
] several ways to generalize theidentities.
O
C
.
h
at 1. Introduction
m
The objective of this paper is to provide a natural combinatorial explana-
[
tionofadivisibilitypropertyoftheKostantpartitionfunction. Thequestion
1 of evaluating Kostant partition functions has been subject of much interest,
v
without a satisfactory combinatorial answer. To mention perhaps the most
8
8 famous such case: it is known that
3 n
n+1
0 K (1,2,...,n,− ) = C ,
. A+n (cid:18) 2 (cid:19) k
1 kY=1
0
1 where C = 1 2k is the Catalan number, yet there is no combinatorial
k k+1 k
1
proof of the abov(cid:0)e i(cid:1)dentity!
:
v Given such lack of understanding of the evaluation of the Kostant parti-
i
X tion function, it seems a worthy proposition to provide a simple explanation
for its certain divisibility properties. We explore divisibility properties of
r
a Kostant partition functions of types A and C , noting that such proper-
n n+1
ties in types B and D are easy consequences of the type C case.
n+1 n+1 n+1
The type A family of identities we study first appeared in a paper by Bal-
n
doniand Vergne [BV], wherethe authors prove the identities usingresidues,
and where they call the divisibility property “mysterious.” It is our hope
that the combinatorial argument we provide successfully demystifies the di-
visibility property of the Kostant partition function and provides a natural
explanation why things happen the way they do.
Theoutline of the paperis as follows. In Section 2 we defineKostant par-
titionfunctionsoftypeA andprovetheBaldoni-Vergneidentitiescombina-
n
torially. Our proof also yields an affirmative answer to a question of Stanley
[S]regardingapossiblebijectiveproofofaspecialcaseoftheBaldoni-Vergne
identities. In Section 3 we define Kostant partition functions of type C ,
n+1
1
2 KAROLAME´SZA´ROS
relate them to flows, and show how to modify our proof for the Baldoni-
Vergne identities to obtain their analogues for type C . We also point out
n+1
other possible variations of these identities in the type C case.
n+1
2. The Baldoni-Vergne identities
Before stating the Baldoni-Vergne identities, we need a few definitions.
ThroughoutthissectionthegraphsGweconsiderareonthevertexset[n+1]
with possible multiple edges, but no loops. Denote by m the multiplicity
ij
of edge (i,j), i < j, in G. To each edge (i,j), i < j, of G, associate the
positive type A root e −e , where e is the ith standard basis vector. Let
n i j i
{{α ,...,α }} be the multiset of vectors corresponding to the multiset of
1 N
edges of G as described above. Note that N = m .
1≤i<j≤n+1 ij
The Kostant partition function K evaluPated at the vector a ∈ Zn+1
G
is defined as
(1) K (a) = #{(b ) | b α = a and b ∈ Z }.
G i i∈[N] i i i ≥0
iX∈[N]
That is, K (a) is the number of ways to write the vector a as a non-
G
negative linear combination of the positive type A roots corresponding to
n
the edges of G, without regard to order. Note that in order for K (a)
G
to be nonzero, the partial sums of the coordinates of a have to satisfy
v +...+v ≥ 0, i∈ [n], and v +...+v = 0.
1 i 1 n+1
We now proceed to state and prove Theorem 1 which first appeared in
[BV]. Baldoni and Vergne gave a proof of it using residues, and called the
result “mysterious.” We provide a natural combinatorial explanation of the
result. Our explanation also answers a question of Stanley in affirmative,
which he posed in [S], regarding a possible bijective proof of a special case
of the Baldoni-Vergne identities.
For brevity, we write G−e, or G−{e ,...,e }, to mean a graph obtained
1 k
from G with edge e, or edges e ,...,e , deleted.
1 k
Theorem 1. ([BV]) Given a connected graph G on the vertex set [n+1]
with m = m = m = 1 and such that
n−1,n n−1,n+1 n,n+1
m +m +m
j,n−1 j,n j,n+1
= c,
m
j,n−1
for some constant c independent of j for j ∈ [n−2], we have that for any
a = (a ,...,a ,− n a ) ∈ Zn+1
1 n i=1 i
P
a +···+a
(2) K (a) = ( 1 n−2 +a +1)K (a).
G c n−1 G−(n−1,n)
Before proceeding to the formal proof of Theorem 1 we outline it, to
fully expose the underlying combinatorics. Rephrasing equation (1), K (a)
G
DEMYSTIFYING THE DIVISIBILITY OF KOSTANT PARTITION FUNCTIONS 3
counts the number of flows f = (b ) on graph G satisfying
G i i∈N
b α = a and b ∈ Z .
i i i ≥0
iX∈[N]
In the proof of Theorem 1 we introduce the concept of partial flows f
H
and the following are the key statements we prove:
• Theelements ofthesetofpartialflowsareinbijection withtheflows
on G−(n−1,n) that the Kostant partition function K (a)
G−(n−1,n)
counts. That is,
# partial flows = K (a).
G−(n−1,n)
• Theelementsofthemultisetofpartialflowsf ,wherethecardinality
H
of the multiset is a1+···+an−2 +a +1 times the cardinality of the
c n−1
set of partial flows, are in bijection with the flows on G that the
Kostant partition function K (a) counts. That is,
G
a +···+a
1 n−2
( +a +1)# partial flows = K (a).
n−1 G
c
The above two statements imply a bijection between the elements of
the multiset of flows on G−(n−1,n) that the Kostant partition function
K (a) counts, where the cardinality of the multiset is a1+···+an−2 +
G−(n−1,n) c
a +1 times the cardinality of the set of flows counted by K (a),
n−1 G−(n−1,n)
and the flows on G that the Kostant partition function K (a) counts, yield-
G
ing
a +···+a
1 n−2
K (a) = ( +a +1)K (a).
G c n−1 G−(n−1,n)
We now proceed to the formal proof of Theorem 1.
Proof of Theorem 1. Let {{α ,...,α }} be the multiset of vectors corre-
1 N
sponding to the edges of G. Let α = e −e ,α = e −e , and
N n−1 n N−1 n−1 n+1
α = e −e . Then equation (2) can be rewritten as
N−2 n n+1
N
#{(b ) | b α = a}=
i i∈[N] i i
Xi=1
N−1
a +···+a
1 n−2
(3) ( +a +1)#{(b ) | b α = a}.
c n−1 i i∈[N−1] i i
Xi=1
Consider a flow f = (b ) , b ∈ Z , of the edges of the graph
H i i∈[N−3] i ≥0
H := G−{(n−1,n),(n−1,n+1),(n,n+1)}. We call f partial if
H
N−3
b α = (a ,...,a ,x ,x ,x ),
i i 1 n−2 1 2 3
Xi=1
4 KAROLAME´SZA´ROS
for some x ,x ,x ∈ Z.
1 2 3
Notice that given a partial flow f = (b ) , it can be extended
H i i∈[N−3]
uniquely to a flow f = (b ) , b ∈ Z , on G−{(n−1,n)}
G−{(n−1,n)} i i∈[N−1] i ≥0
such that N−1b α = a. Furthermore, this correspondence is a bijection.
i=1 i i
Therefore,P
N−1
(4) #{(b ) | b α = a}= 1,
i i∈[N−1] i i
Xi=1 XfH
where the summation runs over all partial flows f .
H
Also, given a partial flow f with Y (f ), i ∈ {n−1,n,n+1}, denoting
H i H
the total inflow into vertex i ∈ {n − 1,n,n + 1} in H, that is the sum
of all the flows b on edges of H incident to i ∈ {n − 1,n,n + 1}, the
i
partial flow f can be extended in Y (f ) + a + 1 ways to a flow
H n−1 H n−1
f = (b ) , b ∈ Z , of G such that N b α = a. Furthermore, given
G i i∈[N] i ≥0 i=1 i i
a flow f = (b ) , b ∈ Z such thaPt N b α = a, there is a unique
G i i∈[N] i ≥0 i=1 i i
partial flow fH = (bi)i∈[N−3] from which itPcan be obtained. Therefore,
N
(5) #{(b ) | b α = a} = (Y (f )+a +1),
i i∈[N] i i n−1 H n−1
Xi=1 XfH
where the summation runs over all partial flows f .
H
Note that since
m +m +m
j,n−1 j,r j,n+1
= c,
m
j,n−1
for some constant c independent of j for j ∈[n−2], it follows that
c Y (f ) = (Y (f )+Y (f )+Y (f )) = (a +···+a ),
n−1 H n−1 H n H n+1 H 1 n−2
Xf Xf Xf
H H H
that is
(a +···+a )
1 n−2
(6) Y (f ) = .
n−1 H
c
Xf Xf
H H
Thus, equation (5) can be rewritten as
DEMYSTIFYING THE DIVISIBILITY OF KOSTANT PARTITION FUNCTIONS 5
N
(a +···+a )
(7) #{(b ) | b α = a}= ( 1 n−2 +a +1)
i i∈[N] i i c n−1
Xi=1 XfH
(a +···+a )
1 n−2
(8) = ( +a +1) 1
n−1
c
Xf
H
N−1
(a +···+a )
1 n−2
(9) = ( +a +1)#{(b ) | b α = a},
c n−1 i i∈[N−1] i i
Xi=1
where the first equality uses equations (5) and (6), and the third equality
uses equation (4). (cid:3)
3. Type C Kostant partition functions and the
n+1
Baldoni-Vergne identities
We now show two generalizations of Theorem 1 in the type C case.
n+1
We first give the necessary definitions and explain the notion of flow in the
context of signed graphs. Throughout this section the graphs G on the
vertex set [n+1] we consider are signed, that is there is a sign ǫ ∈ {+,−}
assigned to each of its edges, with possible multiple edges, and all loops
labeled positive. Denote by (i,j,−) and (i,j,+), i ≤ j, a negative and a
positive edge, respectively. Denote by mǫ the multiplicity of edge (i,j,ǫ) in
ij
G, i ≤ j, ǫ ∈ {+,−}. Toeach edge (i,j,ǫ), i≤ j, of G, associate thepositive
type C root v(i,j,ǫ), where v(i,j,−) = e −e and v(i,j,+) = e +e .
n+1 i j i j
Let {{α ,...,α }} be the multiset of vectors corresponding to the multiset
1 N
of edges of G as described above. Note that N = (m− +m+).
1≤i<j≤n+1 ij ij
The Kostant partition function K evaluatePd at the vector a ∈ Zn+1
G
is defined as
K (a) = #{(b ) | b α = a and b ∈ Z }.
G i i∈[N] i i i ≥0
iX∈[N]
That is, K (a) is the number of ways to write the vector a as a nonnega-
G
tive linear combination of the positive type C roots correspondingto the
n+1
edges of G, without regard to order.
Justlike in the type A case, we would like to think of the vector (b )
n i i∈[N]
as aflow. For this weheregive aprecisedefinition of flows in thetypeC
n+1
case, of which type A is of course a special case.
n
Let G be a signed graph on the vertex set [n + 1]. Let {{e ,...,e }}
1 N
be the multiset of edges of G, and {{α ,...,α }} the multiset of vectors
1 N
corresponding to the multiset of edges of G. Fix an integer vector a =
(a ,...,a ,a ) ∈ Zn+1. A nonnegative integer a-flow f on G is a
1 n n+1 G
vector f = (b ) , b ∈ Z such that for all 1 ≤i ≤ n+1, we have
G i i∈[N] i ≥0
6 KAROLAME´SZA´ROS
(10) b(e)+a = b(e)+ b(e),
v
e∈E,inXc(e,v)=− e∈E,inXc(e,v)=+ e=(Xv,v,+)
where b(e )= b , inc(e,v) = − if edge e= (g,v,−), g < v, and inc(e,v) = +
i i
if e =(g,v,+), g < v, or e = (v,j,ǫ), v < j, and ǫ ∈ {+,−}.
Call b(e) the flow assigned to edge e of G. If the edge e is negative, one
can think of b(e) units of fluid flowing on e from its smaller to its bigger
vertex. If the edge e is positive, then one can think of b(e) units of fluid
flowing away both from e’s smaller and bigger vertex to infinity. Edge e is
then a “leak” taking away 2b(e) units of fluid.
From the above explanation it is clear that if we are given an a-flow f
G
such that b(e) = y, then a = (a ,...,a ,2y − n a ). It is
e=(i,j,+),i≤j 1 n i=1 i
then a matPter of checking the defintions to see that for a signPed graph G on
the vertex set [n+1] and vector a = (a ,...,a ,2y− n a ) ∈ Zn+1, the
1 n i=1 i
number of nonnegative integer a-flows on G is equal toPKG(a).
Thinking of K (a) as the number of nonnegative integer a-flows on G,
G
thereisastraightforwardgeneralization ofTheorem1inthetypeC case:
n+1
Theorem 2. Given a connected signed graph G on the vertex set [n + 1]
with m− = m− = m− = 1, m+ = m+ = m+ = 0, for
n−1,n n−1,n+1 n,n+1 j,n−1 j,n j,n+1
j ∈ [n+1], and such that
m− +m− +m−
j,n−1 j,n j,n+1
= c,
m−
j,n−1
for some constant c independent of j for j ∈ [n−2], we have that for any
a = (a ,...,a ,2y− n a ) ∈Zn+1,
1 n i=1 i
P
a +···+a −2y
1 n−2
(11) K (a) = ( +a +1)K (a).
G c n−1 G−(n−1,n)
The proof of Theorem 2 proceeds analogously to that of Theorem 1.
Namely,definepartialflowsf = (b ) onH := G−{(n−1,n,−),(n−
H i i∈[N−3]
1,n+1,−),(n,n+1,−)} such that
N−3
b α = (a ,...,a ,x ,x ,x ),
i i 1 n−2 1 2 3
Xi=1
for some x ,x ,x ∈ Z and the sum of flows on positive edges is y.
1 2 3
Then, one can prove:
• The elements of the set partial flows are in bijection with the non-
negative integer a-flows on G−(n−1,n). That is,
# partial flows = K (a).
G−(n−1,n)
DEMYSTIFYING THE DIVISIBILITY OF KOSTANT PARTITION FUNCTIONS 7
• Theelementsofthemultisetofpartialflowsf ,wherethecardinality
H
of the multiset is a1+···+an−2−2y +a +1 times the cardinality of
c n−1
the set of partial flows, are in bijection with the nonnegative integer
a-flows on G . That is,
a +···+a −2y
1 n−2
( +a +1)# partial flows = K (a).
n−1 G
c
The above two statements imply a bijection between the elements of the
multisetofnonnegativeintegera-flowsonG−(n−1,n),wherethecardinality
of the multiset is a1+···+an−2−2y +a +1 times the cardinality of the set of
c n−1
nonnegative integer a-flows on G−(n−1,n), and the nonnegative integer
a-flows on G, yielding
a +···+a −2y
1 n−2
K (a) = ( +a +1)K (a).
G c n−1 G−(n−1,n)
Note that the requirement that only negative edges are incident to the
vertices n−1,n,n+1 in G stems from the fact that we need to make sure,
in order for our counting arguments from the proof of Theorem 1 to work,
that we can always assign nonnegative flows to the edges (n−1,n,−),(n−
1,n+1,−),(n,n +1,−) and also, that in case we are extending a partial
flow f to a flow on G, we can extend it in Y (f )+a +1 ways. These
H n−1 H n−1
properties will be satisfied, if we insure that “inflows” at the vertices n−1
andnareatleast−a and−a ,respectively. Tosimplifytheformulation,
n−1 n
we will assume that there are no loops at the vertices n−1,n,n+1, though
the following theorem could also be adopted to a somewhat more general
setting.
Theorem 3. Given a connected signed graph G on the vertex set [n + 1]
with m− = m− = m− = 1, m+ = 0, for i,j ∈ {n−1,n,n+1},
n−1,n n−1,n+1 n,n+1 i,j
and such that
mǫ +mǫ +mǫ
j,n−1 j,r j,n+1
= c,
mǫ
j,n−1
for ǫ ∈ {+,−} and for some constant c independent of j for j ∈ [n−2], we
have that for any a = (a ,...,a ,2y− n a ) ∈ Zn+1, y ≤ a +1,a +1,
1 n i=1 i n−1 n
P
a +···+a −2y
1 n−2
(12) K (a) = ( +a +1)K (a).
G c n−1 G−(n−1,n)
The proof technique of Theorem 3 is analogous to that of Theorem 1. We
invite the reader to check each step of the proof of Theorem 1 and see how
they can be adapted to prove Theorem 3.
8 KAROLAME´SZA´ROS
Acknowledgement
I thank Richard Stanley for his intriguing slides and for pointing out the
work of Baldoni and Vergne. I also thank Alejandro Morales for numerous
conversations about flows and the Kostant partition function.
References
[BV] W.Baldoni,M.Vergne,Kostantpartitionfunctionsandflowpolytopes,inTrans-
formation Groups 13, Birkh¨auser, Boston, 2008, 447-469.
[S] R. Stanley, Acyclic flow polytopes and Kostant’s partition function, Conference
transparencies 2000, http://math.mit.edu/∼rstan/trans.html.
DepartmentofMathematics,MassachusettsInstituteofTechnology,Cam-
bridge, MA 02139
