CC BY
9
УДК 512.542
Kostant Partition Function for sp4(C)
Hasan Refaghat
Department of Mathematics, Islamic Azad University (Tabriz Branch),
Tabriz, Iran
Mohammad Shahryari*
Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz,
Tabriz, Iran
Received 23.07.2011, received in revised form 26.10.2011, accepted 05.11.2011 In this note, we obtain exact values of the partition function of Kostant for the simple Lie algebra sp4(C). Using the values of the partition function, we can find the weight multiplicities of irreducible representations of sp4(C) by a simple computation.
Keywords: symplectic Lie algebra, Kostant partition function, Weyl group, weight multiplicity.
Introduction
Let L be a finite dimensional complex semi-simple Lie algebra with a Cartan subalgebra H and root system $ and suppose $+ denotes the set of positive roots. Recall that the Weyl vector is defined by
p = Y^ a
Suppose A is an integral dominant weight of L and V(A) is the corresponding irreducible L-module. For any other integral dominant weight we define the multiplicity of ^ in A to be the dimension of
V(A)M = {v e V(A) : Vh e H h.v = /j,(h)v}.
There exists a compact formula for computing weight multiplicities, known as Kostant's multiplicity formula. It can be stated as follows,
dim V(A)M = £ eMPMA + p) - (M + p)).
wew
Here W is the Weyl group of L, e(w) is the sign of w, and P is the Kostant partition function. By definition, for any weight 7, P(y) is the number of ways to write 7 as a linear combination of positive roots with non-negative coefficients, (see [1] for details). Although the Kostant partition function is a well-known classical notion in Lie algebra, an explicit expressions for it, might not be easy to find in the literature. In this note, we give an explicit formula for the values of the partition function, in the case L = sp4 (C).
*[email protected] © Siberian Federal University. All rights reserved
1. Generalities
In this elementary section, we give a review of theories concerning the Lie algebra sp4(C). We also fix a set of notations we will use in the next sections.
The symplectic Lie algebra, sp4(C), is a 10-dimensional simple Lie algebra defined by
sp4(C) = {x e Mat4(C) : sx = -xTs},
where
0 I2 I2 0
A Cartan subalgebra for sp4(C) is H which consists of diagonal matrices
h = diag(a\, a2, —ai, —a2), where ai, a2 e C. For i = 1, 2, define a functional Mi : H ^ C by ^i(h) = aj. So the set
$ = {±mi ± Mj : 1 < i,j < 2} —{0} is a root system for sp4(C). Also, the set
n = {Ri = Mi — M2, R2 = 2^2}
is a basis for Finally
Ai = Mi, A2 = Mi + M2 are fundamental weights of sp4(C). Note that, we also have the following simple relations;
Mi = Ai
M2 = —Ai + A2
Ri = 2Ai — A2
R2 = —2Ai + 2A2
Ai = Ri + 1R2 A2 = Ri + R2 •
We denote the Weyl group by W. It is generated by the reflection aa, a e n, where
(P) P 2Q3,a)
aa(p) = p — --- a,
( a, a)
for p e Let ai = aRl and a2 = . In the following table, we give the elements od W by their actions on the elements m1 and m2, their expressions as products of ai and a2, and their sign, e(w).
The Weyl group of sp4(C)
w(M2) presentation e(w)
Mi M2 1 1
W2 M2 Mi
W3 Mi -M2
w4 M2 -Mi aia2 1
W5 -M2 Mi 1
W6 -Mi M2
W7 -M2 -Mi
W8 -Mi M2 (^i ^2 )2 1
Let A = aAi + bA2 be an integral dominant weight of sp4(C). We denote the corresponding sp4(C)-module by V(A). Using Weyl's dimension formula, (see [1], page 267), we have
dim V (A) = ^(a + 1)(b + l)(a + b + 2)(a + 2b + 3). 6
Equivalently, if A = + q^2, we have
dim V(A) = 6(p + 2)(q +1)(p + q + 3)(p - q + l).
Now, we are going to compute the weight multiplicities for sp4(C). These are very important for decomposition of a sp4 (C)-module in to the direct sum of irreducible constituents. For this purpose, we use the Kostant multiplicity formula;
dim V(A)M eMPMA + p) - + p)),
/m
wew
where p = A1 + A2 and P is the partition function, i.e. P(^) is the number of ways to write ^ as a linear combination of positive roots with non-negative integer coefficients.
Suppose A = + q^2 and ^ = + s^2. Since we have p = 2^1 + so using the above table,
8
dim V(A)M = ]T e(c*)P(wj(A + p) - (M - p))
i=1 8
= ]T e(c*)PM(p + 2)M1 + (q +1)M2)
i=1
-((r + 2)^1 + (s +1)^2)) = p((p - r)^1 + (q - s)^2)
-P((q - r - 1)M1 + (p - s + 1)M2) -P((p - r)M1 - (q + s + 2)M2) +P((q - r - 1)M1 + (p + s + 3)M2).
It is enough to know the values of P(y) for the following cases, Case 1: 7 = i^ + j^2 such that i, j > 0 and i + j is even. Case 2: 7 = - j^2 such that i > j > 0 and i + j is even.
2. Values of the Partition Function
In what follows, we obtain the exact values of P(y). We know that the positive roots of sp4(C) are
A1 = M1 - M2, A2 = M1 + M2, = 2^1, $4 = 2^2.
Now, we have
4
P(y) = |{(r1,r2,rs,r4) : r G Z,r > 0,7 = ^3riA}|.
i=1
Let 7 = + j^2. If it is possible to write 7 as non-negative integer linear combination of $js, then we have
+ j>2 = r1$1 + r2$2 + r3$3 + r4$4
= (r1 + r2 + 2r3)^1 + ( r 1 + r2 +2r4)^2,
so we must have
i = ri + r2 + 2r3, j = -ri + r2 + 2r4. Hence we obtain the the following results;
1. If i + j is odd, then P(y) = 0.
2. If i < 0, then P(y) = 0.
3. If -j > i > 0, then P(y) = 0.
It is enough to know the values of P(^) for the following cases,
Case 1: y = + j^2 such that i, j > 0 and i + j is even.
Case 2: 7 = — j^2 such that i > j > 0 and i + j is even.
Let 7 = +j^2 such that i, j > 0 and i+j is even. Then P(y) is the number of non-negative integer solutions of
ri + r2 + 2r3 = i -ri + r2 + 2r4 = j
Adding up two equations, we obtain
i+j
r2 + r3 + r4 = ——.
Also, subtraction gives
i — j , ri = —2--+ r4 — r3.
So, P(y) is the number of integer solutions of the following system;
i+j
x + y + z = —2— x + 2y ^ i x + 2z ^ j x, y, z ^ 0
We consider two cases: j < i and i < j. In the first case, we see that P(y) is the number of solutions of the system;
. i + j x + y < —2~
x + 2y ^ i x, y ^ 0
Lemma 2.1. The number of integer solutions of the system
x + y ^ k x, y ^ 0
is equal to
(k +1)(k + 2) n = ---.
Proof. For 0 < l < k, let
Ai = {(x, l) : x > 0, x + l < k}. So, we have |Ai | = k - l + 1. Thus
n = ¿|Ai|=(k + 1)<k + 2).
i=1
Corollary 2.1. The number of the integer solutions of the system
x + y < k
0 < y < 1
x > 0
is equal to
(l + 1)(2k + 2 - l) 2 '
Lemma 2.2. The number of the integer solutions of the system
x + 2y ^ k x, y ^ 0
is equal to
(k + 2)2
n :
4
if k is even, otherwise
(k +1)(k + 3)
n = -.
4
Proof. For 0 < / < k, suppose
Ai = {(x, /) : x > 0, x + 2/ < k}.
Then, we have
|Ai| = k - 2/ +1,
and hence, for even k
n =1 + 3+ ••• + (k +l) = ^+2)2,
and for odd k, we have
N (k +1)(k + 3) n = 2 + 4+ ••• + (k + 1)= ( + )( + ).
Corollary 2.2. Suppose n is the number of integer solutions of the system
x + 2y ^ k 0 < / < y x ^ 0.
If k is even, then we have
and otherwise
(k - 21 + 2)2 n =-4-'
(k - 21 +1)(k - 21 + 3)
We return to the system
x + y <
i + j
x + 2y ^ i x, y ^ 0
where j < i and i + j is even. One can split this system into two complementary systems as follows,
x + y <
0 < y <
i + j
i - j
and
x ^ 0
x + 2y ^ i
i — j
x > 0.
So, using Lemmas 3-2 and 3-4, we obtain
Corollary 2.3. Let y = i^i + j^,2 such that 0 ^ j < i and i + j is even. If j is even, then
(i — j +2)(i + 3 j +4) + 2j2
P(Y)
8
and if j is odd, then we have
P(y)
(i - j + 2)(i + 3j + 4) + 2(j2 - 1) 8 '
Now, suppose we have i < j. It is easy to see that in this case, P(y) is the number of solutions of the system
x + 2y ^ i x,y ^ 0.
Hence, we obtain;
Corollary 2.4. Let 7 = + j^2 such that 0 ^ i ^ j and i + j is even. If i is even, then
(i + 2)2
P(Y)
4
while in the other case
p(y ) =
(i + 1)(i + 3)
n =
4
2
2
2
Finally, we consider the case 7 = i^ — j^2, with i + j even and 0 < j < i. Again, P(y) is the number of non-negative integer solutions of the system
ri + r2 + 2r3 = i —ri + r2 + 2r4 = —j
Adding up two equations, we obtain
i — j
Also, subtraction gives
We see that if (r2, r3, r4) satisfies
Г 2 + Г3 + Г4 =
« + j . ri = —--+ Г4 - Г3.
Г2 + Г3 + Г4
« - j
Г2 > 0 Г3 > 0 Г4 > 0
then (ri,r2,r3,r4) is a non-negative integral solution to the above system, and moreover every solution can be obtain by this method. Hence the required number is just the number of solutions of the simple sysytem
i — j
x+у < x, у ^ 0
and so we have
Corollary 2.5. Suppose 7 = i^i — with i + j even and 0 < j ^ i.Then we have
P(Y) = (i — j + 2f — j + 4).
References
[1] R.W.Carter, Lie algebras of finite and affine type, Cambridge University Press, 2005.
Функция разбиения Костанта для sp4(C)
Хасан Рефагат Мохаммад Шахрири
В этой статье получены точные значения функции 'разбиения Костанта для простой алгебры Ли sp4(C). Используя значения функции разбиения простыми вычислениями найдены весовые кратности неприводимых представлений sp4 (C).
Ключевые слова: симплектическая алгебра Ли, функция разбиения Костанта, группа Вейля, весовая кратность.
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