CC BY
6
Journal of Siberian Federal University. Mathematics & Physics 2017, 10(4), 494-502
YflK 517.9
Weight q-multiplicities for Representations of sp4(C)
Pamela E. Harris* Edward L. Lauber^
*
Williams College Williamstown, MA 01267
USA
Received 18.01.2017, received in revised form 03.04.2017, accepted 20.08.2017
In this paper we present a closed formula for the values of the q-analog of Kostant's partition function for the Lie algebra sp4(C) and use this result to give a simple formula for the q-multiplicity of a weight in the representations of the Lie algebra sp4(C). This generalizes the 2012 work of Refaghat and Shahryari that presented a closed formula for weight multiplicities in representations of the Lie algebra sp4(C).
Keywords: Sympletic Lie algebra, Kostant partition functions, q-analog of Kostant partition function, weight multiplicity, weight q-multiplicity. DOI: 10.17516/1997-1397-2017-10-4-494-502.
1. Introduction and preliminaries
In the study of the representation theory of simple Lie algebras it is of interest to compute the multiplicity of a weight ^ in a finite-dimensional complex irreducible representation of the Lie algebra g of rank r. This multiplicity is the dimension of a specific vector subspace called a weight space. The theorem of the highest weight asserts that all finite-dimensional complex irreducible representation of g are equivalent to L(A) a highest weight representation with dominant integral highest weight A = n1m1 + n2m2 + • • • + nr mr where ni, n2,... ,nr G N := {0,1, 2,...} and {m1, m2,... ,mr } denotes the set of fundamental weights of g. One can compute this multiplicity, denoted by m(A,^), by using Kostant's weight multiplicity formula [10]:
where W is the Weyl group, 1(a) denotes the length of a, p = — ^ a with being the set of
2 a£$+
positive roots of g. If h is a Cartan subalgebra of g, then elements of the dual of h, denoted by h*, are called weights. In Equations (1) p denotes Kostant's partition function defined on weights £ G h* and p(£) counts the number of ways the weight £ can be expressed as a nonnegative integral sum of positive roots.
A generalization of Equations (1) was provided by Lusztig and is called the ^-analog of Kostant's weight multiplicity formula [11]:
(1)
aew
1
aew
* [email protected] [email protected] © Siberian Federal University. All rights reserved
with pq denoting the q-analog of Kostant's partition function defined on £ e h* by
Pq (£) = Co + ciq + c2q2 +-----+ cnqn,
where Ci represents the number of ways to express the weight £ as a sum of exactly i positive roots. Note that since pq(£)|q=i = p(£) for all weights £ e h* evaluating mq(A, j) at q = 1 recovers m(A,j). The following is a celebrated result of Lusztig, which illustrates an important use of the q-analog of Kostant's weight multiplicity formula [11, Section 10, p. 226]: if g is a finite-dimensional simple Lie algebra, then mq (a, 0) = qei + qe2 + • • • + qer where a is the highest root and e1,e2,... ,er are the exponents of g. We recall that the exponents of g are related to the degrees of the basic invariants, where the degrees are equal to one more than the exponents [9].
Even though formulas, such as Equations (1) and (2), exist to compute the multiplicity and q-multiplicity of the weight j in the irreducible representation L(A), respectively, the computation can be intractable. This is due to the fact that in general the number of terms appearing in the sum are factorial in the rank of the Lie algebra and there is no known closed formula for the partition function involved. There has been recent progress in addressing these complications for particular weight multiplicity computations [4, 5, 7, 8].
Although it is very difficult to give closed formulas for weight multiplicities in rank r Lie algebras, there has been some success in low rank cases. One such case is the work of Refaghat and Shahryari, where they provided a closed formula for Equation (1) for the Lie algebra sp4(C) [12]. More recently, Fernandez-Nunez, Garcia-Fuertes, and Perelomovto provided a generating function for the weight multiplicities of the representations of the Lie algebra sp4(C) [1]. Motivated by the work of Refaghat and Shahryari and of Fernandez-Nunez, Garcia-Fuertes, and Perelomovto we present a new formula for Equation (2) which gives the q-multiplicities for weights of the representations of the Lie algebra sp4 (C). This formula depends solely on a simple computation involving the values of A and j and not on the partition function nor on the order of the group.
Theorem 1.1. Let w1 and w2 denote the fundamental weights of sp4(C) and consider A =
^^ — x
mwi + nm<2 and j = xwi + with m, n,x,y e N. Define a = m+n-x — y, b = n-y +--,
c = n — x — y — 1, and d = —y — 1 H--. Then
P — Q — R if a,b,c,d& N,
P — Q
mq(A, ¡j) = \ P — R P 0
if a,b,cG N and N, if a,b,d£ N and N, if a,b£ N and c,d <// N, otherwise,
(3)
where
P
n(La \'b*> / a+b-2i \ 12 \ /b+c-2i \ d /a+d-2i
E ( E ) , Q = E ( E ) , and R=£ ( £ qj
i=0 \j=max(a-i,b) J i=0 V j=b I i=0 V j=a-i
The proof of Theorem 1.1 uses the following closed formula for the q-analog of Kostant's partition function.
Proposition 1.2. Let ai and a2 denote the simple roots of sp4(C). If m,n£ N, then
min([m \,n) f m+n-2i
pq (mai + na2)
E
i=0
E
(4)
\j=m&x(m-i,n)
j
q
Equation (3) yields a generalization of the formula for Kostant's weight multiplicity formula for sp4(C) presented in [12] as its evaluation at q =1 recovers their result. This paper is organized as follows: Section 2 provides the necessary background to make our approach precise. Section 3 provides a proof of Proposition 1.2. Lastly, Section 4 contains the proof of Theorem 1.1 and Corollary 4.1 which considers the case of setting q =1 in Theorem 1.1 thereby giving a closed formula for Kostant's weight multiplicity for the Lie algebra sp4(C).
2. Background
Following the notation of [2, 3] we now provide the necessary background to make our approach precise. Throughout this work we let ai,a2 denote the simple roots and m1,m2 the fundamental weights of sp4(C). One may change from fundamental weights to simple roots via
1
mi = ai + - a2, (5)
2
w2 = ai + a2.
(6)
We consider the case where A = mm1 + nm2 and j = xm1 + ym2 with m, n,x,y G N, thereby using the fundamental weights as an initial basis for A and j, but we often convert to the simple roots in order to simplify partition function calculations.
The set of positive roots of sp4(C) is given by = {a1; a2, a1 + a2, 2a1 + a2} and hence
1 ^ 3
P = 2 a = 2a i + - a2 = w + W2.
(7)
The Weyl group of sp4 (C) is denoted by W and its elements are generated by the root reflections si and s2, which are perpendicular to the simple roots a1 and a2, respectively. The eight elements of W and their lengths are presented in Tab. 1. The action of s1 and s2 on the simple roots and
Table 1. The elements of the Weyl group of sp4 (C) and their length
a G W 1 «1 «2 S1S2 «2 Si «1«2«1 «2«1«2 «1 «2«1«2
£(a) 0 1 1 2 2 3 3 4
the fundamental weights is given by
si(ai) = -ai, S2(ai) = ai + a2, si(a2) = 2ai + a2, S2(a2 ) = -a2
and for 1 ^ i,j ^ 2
i{Wj ) =
if i = j, if i = j.
(8)
(9)
The action of any other element of the Weyl group is acquired by noting that the action of si
and s2 is linear. For example, sis2(3m2) = 3si(s2(m2)) = 3si(m2 — a2) = 3(si(m2) — si(a2)) = 3(m2 — (2ai + a2)) = 3((ai + a2) — (2ai + a2)) = —3ai.
3. The q-analog of Kostant's partition function
The following sections consider a weight £ and analyze the value of pq (£) when using the positive roots of the Lie algebra sp4(C). In this analysis we note that combinatorially the
w
3
Wo — a
3
3
positive roots of the Lie algebra of sp4(C), = {a1, a2, a1 + a2, 2a1 + a2} all but 2a1 + a2 are positive roots of the Lie algebra s(3(C), whose positive roots are ai, a2, and ai + a2. Hence, we first present a closed formula for the q-analog of the partition function for the Lie algebra s(3 (C) and use this result in our proof for the closed formula for the q-analog of the partition function for the Lie algebra sp4(C).
3.1. Formula for pq on 5(3 (C)
By examining the partition function value on the Lie algebra s(3(C), we can begin to better understand the partition function value on sp4(C). We consider the q-analog of Kostant's partition function on ma1 + na2 with m,n G N in the Lie algebra 5(3 (C).
m+n
Proposition 3.1. If g = s(3(C) and m,n G N, then pq(ma1 + na2) = ^ qj.
j=max(m,n)
Proof. The number of possible ways to write ma1 + na2 as a nonnegative integral sum of the positive roots a1, a2, a1 + a2 is determined entirely by the number of times a1 + a2 is used. If a partition of ma1 + na2 includes c(a1 + a2), where 0 ^ c ^ min(m, n), then there must be m — c uses of a1 and n — c uses of a2. The total number of roots used in this partition will be m + n — c. Since we know that c ranges in value from 0 to min(m,n), and that there is one and only one possible partition for each value of c, it follows that the number of roots used in a partition of ma1 + na2 ranges between m + n — min(m, n) = max(m, n) and m + n. Thus,
m+n
pq (mai + na2) = qj. 1=1
j=max(m,n)
The next corollary follows directly from Proposition 3.1. Corollary 3.2. Let g = s(3(C). If m,n G N, then p(ma1 + na2) = min(m, n) + 1. Proof. We note that pq(ma1 + na2)|q=1 = p(ma1 + na2). Thus, we set q = 1 and find that
m+n
m+n
E
max(m,n)
= ^^ 1 = m + n — (max(m, n) — 1) = min(m, n) + 1. □
q=1 j=max(m,n)
3.2. Formula for pq on sp4(C)
We now consider the Lie algebra sp4(C) with positive roots ai, a2, ai + a2, 2ai + a2.
Proof of Proposition 1.2. We note that every partition of a weight that is possible in st3(C) is also possible in sp4(C). However, sp4 (C) also has 2a1 + a2 as a positive root, so we must consider all partitions of a weight using this root. Let m,n G N. It is clear that any partition of ma1 + na2 can contain i copies of the positive root 2a1 + a2 so long as 0 < i < min^,n). It follows that when using 0 ^ i ^ min([,n) copies of the root 2a1 + a2 to partition ma1 + na2, the remainder (m — 2i)a1 + (n — i)a2 must be partitioned using only the roots a1, a2, a1 + a2. Thus, by Proposition 3.1 the number of ways to partition (m — 2i)a1 + (n — i)a2 using only the positive roots a1, a2, a1 + a2 is given by
m+n-3i
E ■ (1°)
j=max(m- 2i,n-i)
In our count we must add i to the exponents of q in every term of expression (10) to account for the i copies of the root 2ai + a2 used in the partition of mai + na2. Doing this yields the polynomial
qmax(m-i,n) + ^max(m-i,n) + 1 + + ^m+n-2i
By accounting for the fact that 0 < i < min( [mm J , n) we arrive at the desire result
min([ f J,n) / m+n-2i pq (mai + na2) = ^ I qj
i=0 \j=max(m—i,n)
We now obtain a closed formula for Kostant's partition function on sp4(C). Corollary 3.3. If g = sp4(C) and m,n G N, then
+ 1)
□
+ 1 m -
p(ma1 + na2) = <
m
L~2-
(n + 1)(n + 2)
, 2 2 2 2mn - m2 - n2 + m + n
+
m ( m
- m - —
L 2 J L 2 J
if n ^ m, if m ^ 2n, + 1 if 2n > m > n.
Proof. Setting q =1 into Equation (4) we find that
min(m — i, n)^ —
p(ma1 + n«2)
' min( [\ ,n)
E
i=0
1 . ( m \ ( • ( m
- min — , n min —
2 V L 2 J 'A V L 2 J
m
L~2
We now consider each case individually. If n > m, then Equation (11) simplifies to
2
+ 1 m -
2
+ . If m ^ 2n, then Equation (11) simplifies to
(n + 1)(n + 2)
. Fi-
nally, if 2n > m > n, then Equation (11) yields
([ 2 \ E
min(m - i, n) I - L^J ( L^J +1)
+
i=0
m
~2J
+ 1.
(12)
Let us consider the first term of expression (12). If i ^ m — n, then n ^ m — i and hence min(m — i, n) = n. Similarly, if i > m — n, then n > m — i and hence min(m — i, n) = m — i. Thus
EU \ . 2mn — m2 — n2 + m + n
min(m — i, n) =--h m
i=0 2
m
L~2 J
L?J( L?J +1)
Substituting Equation (13) into Equation (12) yields the desired result.
(13) □
4. The q-analog of Kostant's weight multiplicity formula for sp4(C)
Proposition 1.2 provided a closed formula for the q-analog of the partition function for the Lie algebra sp4(C). We now use this formula to provide a proof of Theorem 1.1, but first we
n
n
m
m
2
2
must examine the partition function value input a(A + p) — (p + p) for all a G W as appearing in Equation (1). Throughout the rest of this section we let A = mw\ + nw2 and p = xwi + yw2 with m, n,x,y G N. To illustrate the computations a(A + p) — (p + p) we consider the case when a = si. Using Equations (5), (6), and (8) we find that
si(A + p) — (p + p) =
=si ^(m + n + 2)ai + ^y + n +0 a^ — ^(x + y + 2)ai + ^X + y + ^
= (n + 1)ai + ^22 + n +2) a2 — ^(x + y + 2)ai + ^X + y + ^ a2^j
m—x
= (n — x — y — 1)ai + n — y +--2— ) a2
Repeating this procedure with every element of the Weyl group generates the content of Tab. 2.
Table 2. Computing a(A + p) — (p + p) for all a G W
a 1(a) a(A + p) — (p + p)
1 0 (m + n — x — y)ai + (n — y + mp )a2
si 1 (n — x y 1)ai + (n y + m,2x )a2
S2 1 (m + n x y)ai + ( y 1 + m,2x )a2
SiS2 2 (—n — x — y — 3)ai + (—y — 1 + mp )a2
S2Si 2 (n — x -y — 1)ai + (—y — 2 — mp )a2
SiS2Si 3 (—m — n — x — y — 4)ai + (— y — 2 — )a.2
S2SiS2 3 (—n — x — y — 3)ai + (— n — y — 3 — ^ )a2
SiS2SiS2 4 (—m — n — x — y — 4)ai + (—n — y — 3 — )a2
We now consider the ^-analog of Kostant's partition function on the expressions a(A + p) — (p + p) as listed in Tab. 2. We note that m,n,x,y G N and that the ^-analog of Kostant's partition function returns 0 if the coefficient of either ai or a2 is negative or not an integer. Thus, the Weyl group elements sis2, s2si, sis2si, s2sis2, and sis2sis2 never contribute to the ^-analog of Kostant's weight multiplicity formula since at least one of the coefficients of ai or a2 in the expression a(A + p) — (p + p) will always be negative whenever A = mmi + nw2 and p = xwi + yw2 with m, n,x,y G N. With this observation at hand we now present the proof of our main result Theorem 1.1.
Proof of Theorem 1.1. We need only consider the contribution of ( — 1)l(a)pq(a(A + p) — (p + p))
mx
for the Weyl group elements a = 1, si, and s2. Let a = m + n — x — y, b = n — y +--2—, c =
mx
n — x — y — 1, and d = —y — 1 +--. Then from Tab. 2 we note that
1(A + p) — (p + p) = aai + ba2, si(A + p) — (p + p) = cai + ba.2, S2 (A + p) — (p + p) = aai + d,a.2.
By Proposition 1.2 if a = 1 and a,b G N, then
min(L f \'b) / a+b-2i P = ( — 1)£(1)pq (aa1 + ba2)= J2 I ]T
q
(14)
= 0 \j=max(a-i,b)
Observe that if a G N or b G N, then pq(aa1 + ba2) = 0 and hence P = 0. Note that if m, n,x,y G N, then
- x - y — 1 < y — -
n — x — y — 1 < n — y +
c <b.
Thus, Proposition 1.2 allows us to compute the following: If a = s1 and b,c G N with c <b, then
LfJ /b+c
g = (—1)e(si)p(cai + ba2) = — £ ¡J2qj I.
i=0 \j=b
(15)
However, if b G N or c G N, then pq(ca1 + ba2) = 0 and hence Q = 0. Finally, since m, n,x,y G N, we have
n — y n — y + m — x m — x a
> —y — 1 ^ 2- > —y — 1 + — ^ 2 >d
Thus, Proposition 1.2 allows us to compute the following: If a = s2 and if a,d G N with a > 2d, then
d l a+d-2i
R =( — 1)e(s2)p(aai + da2) = — ^ I ^
q
(16)
i=0 \j=a-i
Again, if a G N or d G N, then pq(aa1 + da2) = 0 and hence R = 0. Equation (3) now follows from taking the sum of Equations (14)-(16). □
Our last result follows from setting q =1 in Equation (3) of Theorem 1.1 and using Corollary 3.3.
Corollary 4.1. Let X = mrni + nw2 and ¡j, = xwi + yw2 with m,n,x,y G N := {0,1, 2,...} be
mx
weights of sp4(C) and define a = m + n — x — y, b = n — y +--2—, c = n — x — y — 1, and
m — x
d = —y — 1 +--2—. Then
'P — Q — R if a, b,c, d G N,
m(A, ¡j) = <
p — g
P — R
P
0
if a,b,cG N and d / N, if a,b,d£ N and c / N, if a,b G N and c,d / N, otherwise,
(17)
where
(a — LU+0
P =
,L2J+1J la — (b + 1)(b + 2)
2
2ab — a2 — b2 + a + b
+
if b > a, if a ^ 2b, ) + 1 if 2b > a > b,
g
R
c + 2 2
(d +1)(d + 2)
2 '
mx
a
We end by providing a computational proof that mq(a, 0) = qi + q3 where a is the highest root and 1 and 3 are the exponents of sp4(C).
Example 1. Let A = a = 2m i and p = 0. Then m = 2, n = x = y = 0, and a = m+n—x—y = 2,
m x m x b = n — y +--= 1, c = n — x — y — 1 = —1, and d = —y — 1 +--= 0. By Theorem 1.1
as a,b,d G N and c G N we must compute P and R. Observe that
it 3-2i \ 0 / 2 —2i \
P = E I E qj) = q + q2 + q3 and r= £ I £ qj I = q2.
i=0 yj=max(2— i, i) J i=0 \ j=2 J
Thus mq(a, 0) = P — R = q + q3. Lastly, note that mq(a, 0)|q=i = 2 recovers the rank of the Lie algebra sp4(C).
The authors thank Gabriel Ngwe for edits to an earlier draft of this manuscript and Leo Goldmakher for his assistance in translating the title and abstract to Russian.
References
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Вес q-кратностей для представлений sp4(C)
Памела Е. Харрис Эдвард Л. Лаубер
Вильямс колледж Вильямстаун, MA 01267 США
В настоящей 'работе мы приводим замкнутую формулу для значений q-аналога функции обобщения Костанта для алгебры Ли sp4(C) и используем этот результат, чтобы дать простую формулу для q-кратности веса в представлениях алгебры Ли sp4(C). Это обобщает работу Рефагата и Шахрияри в 2012 г., которые дали замкнутую формулу для кратности веса в представлениях алгебры Ли sp4(C) .
Ключевые слова: симплектическая алгебра Ли, статистическая сумма Константа, q-аналог статистической суммы Константа, кратность веса, вес q-кратности.
