Computation of Central Elements

Computation of Central Elements

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 15 October 2010

Computation of ZV(l) when V=L ε1

Let 𝔤=𝔰𝔬2r+1 or 𝔤=𝔰𝔭2r or 𝔤=𝔰𝔬2r, and let V=L ε1 , y= ε1, ε1+2ρ ε= 1, if  𝔤= 𝔰𝔬2r+1, -1, if  𝔤= 𝔰𝔭2r, 1, if  𝔤= 𝔰𝔬2r, δ= 0, if 𝔤 =𝔰𝔬2r+1, 1, if 𝔤 =𝔰𝔭2r, -1, if 𝔤 =𝔰𝔬2r. Let zV(l) be the central elements in Uq𝔤 defined by zV(l) = ε idtrV t+ 12 y l and let zV u = i0 zV(l) u-l. Then π0 zV u + ε u - 12 = εu + 12 u+ 12 y u- 12 y -1 u- 12 δ u- 12 y u+ 12 y +1 u+ 12 δ σρ i=1 r u + hi + 12 u- hi+ 12 u+ hi - 12 u- hi - 12 .

Proof.
By (5.20), as operators on L0V, t = 12 ε1 , ε1 + 2ρ - ε1 , ε1 + 2ρ + 0, 0+ 2ρ =0 and 12 y +t l = 12 y l . So, as operators on L0V, zV(l) = ε idtrV 12 y + t l = ε idtrV 12 y l = ε dim V 12 y l and, since dimV=ε+y, zV u = l0 zV(l) u-l = l0 ε dim V 12 y l u-l = ε dimV 1 1- 12 y u-1 . Therefore
zV u + εu - 12 = 1+εy 1- 12 y u-1 +εu - 12 = u+εyu+ εu - 12 u- 12 y u- 12 y = ε u 2 + 12 u + 12 ε y u + 14 y u- 12 y = εu + 12 u+ 12 y u- 12 y . 6.13
By the second identity in (2.6) and the definition of Φ in Theorem 5.2(a), yk𝒲k-1 acts on L0 Vk = L0 Vk-1 V as 12 y + t. If Lμ is and irreducible Uq𝔤-module in L0Vk-1, then (5.20) gives that yk acts on the Lλ component of LμV by the constant
cλ,μ = 12 y + 12 μ±εk, μ±εk +2ρ - μ, μ+2ρ - ε1, ε1+2ρ = 12 y + 12 ±2μi ±2ρi +1 -y = ±μi ± 12 y-2i+1 + 12 = ± 12 + 12 y +μi -i + 12 = - 12 + 12 y + λi + 12 if μλ, - 12 - 12 y - μi -i + 12 , if μλ, = 12 y + λi -i, if μλ, - 12 - μi-i , if μλ, = 12 y + cλ/μ, if μλ, - 12 y - cμ/λ if μλ. 6.14
As in [
Naz, Theorem 2.6], the irreducible 𝒲k-module 𝒲 k μ/0 = 𝒲 k μ has a basis vT indexed by up-down tableaux T = T(0), T(1), , T(k) , where T(0)=, T(k)=μ, and T(i) is a partition obtained from T(i-1) by adding or removing a box, and yi vT = 12 y + cb vT, if  b=T(i) / T(i-1), 12 y - cb vT, if  b= T(i-1) / T(i). Thus i=1 k-1 u-yi-1 u+yi+1 u-yi 2 u+yi 2 u-yi+1 u-yi-1 acts on the Lμ 𝒲 k-1 μ isotypic component in the Uq𝔤𝒲k-1-module decomposition
L0 Vk-1 μ Lμ 𝒲 k-1 μ , 6.15
by
i=1 k-1 u+ c T(i) , T(i-1) -1 u+ c T(i), T(i-1) +1 u-c T(i), T(i-1) 2 u+c T(i), T(i-1) 2 u-c T(i), T(i-1) +1 u-c T(i), T(i-1) -1 6.16
for any up-down tableaux T of length k and shape μ. If a box is added (or removed) at step i and then removed (or added) at step j, then the i and j factora of this product cancel. Therefore, (
6.16) is equal to
bμ u+ 12 y +cb -1 u+ 12 y +c b +1 u- 12 y -c b 2 u+ 12 y +c b 2 u- 12 y +c b +1 u- 12 y -c b -1 6.17
(see [
Naz, Lemma 3.8]). Simplifying one row at a time,
bμ u+ 12 y +c b -1 u+ 12 y +c b +1 u+ 12 y +c b 2 = i=1 r u+ 12 y -i u+ 12 y+ μi -i+1 u+ 12 y +1 -i u+ 12 y + μi -i = u+ 12 y -r u+ 12 y +1 i=1 r u+ 12 y + μi -i +1 u+ 12 y + μi -i , 6.18
if μ=μ1μr. It follows that (
6.17) is equal to
u+ 12 y -r u+ 12 y +1 u- 12 y -1 u- 12 y +r i=1 r u+ 12 y + μi -i +1 u+ 12 y + μi -i u- 12 y - μi-i u+ 12 y - μi -i +1 = u- 12 δ u+ 12 y +1 u- 12 y -1 u+ 12 δ evμ+ρ i=1 r u+ hi+ 12 u+ hi - 12 u- hi + 12 u- hi - 12 6.19
where δ = 0, if 𝔤= 𝔰𝔬2r+1, 1, if 𝔤= 𝔰𝔭2r, -1, if 𝔤= 𝔰𝔬2r, since evμ σρ h ε i = evμ+ρ h ε i = μ+ρ, ε i = μi + 12 y-2i+1 = 12 y + 12 -i and 12 y -r = 12 y-2r = 12 y-2r+δ - 12 δ = - 12 δ Combining (
6.13) and (6.19), the identity (4.14) gives that zV u + εu - 12 = zV 1 + εu - 12 i=1 k-1 u+ yi -1 u+ yi +1 u- yi 2 u+ yi 2 u- yi +1 u- yi -1 acts on the L μ 𝒲 k-1 μ -component in the Uq𝔤 𝒲k-1 -module decomposition in (6.15) by εu + 12 u+ 12 y u- 12 y u- 12 δ u+ 12 y +1 u- 12 y -1 u+ 12 δ evμ+ρ i=1 r u+ hi+ 12 u+ hi - 12 u- hi + 12 u- hi - 12 .

To help illuminate the cancellation done in (6.18), we walk through the example where μ= 553311 , and so the contents of the boxes are TABLEAU GOES HERE. In this example, the product over the boxes in the first row of the diagram is b in first row of μ x+cb -1 x+cb +1 x+cb x+cb = x-1 x+1 x+0 x-0 x+0 x+2 x+1 x+1 x+1 x+3 x+1 x+2 x+2 x+4 x+3 x+3 x+3 x+5 x+4 x+4 = x-1 x+0 x+5 x+4 , where   x=y+ 12 y. Thus, simplifying the product one row at a time, bμ x+cb -1 x+cb +1 x+cb x+cb = x-1 x+0 x+5 x+4 x-2 x-1 x+4 x+3 x-3 x-2 x+1 x+0 x-4 x-3 x+0 x-1 x-5 x-4 x-3 x-4 x-6 x-5 x-4 x-5 = x-6 x+0 x+5 x+4 x+4 x+3 x+1 x+0 x+0 x-1 x-3 x-4 x-4 x-5 leading us to the identity bμ x+cb-1 x+cb+1 x+cb x+cb = x-r x+0 i=0 r x- μi -i +1 x+ μi -i , if   μ= μ1 μr .

Let 𝔤=𝔰𝔬2r+1 or 𝔤=𝔰𝔭2r or 𝔤=𝔰𝔬2r, and let V=L ε1 and z=εqy, where y= ε1, ε1+2ρ ε= 1, if  𝔤= 𝔰𝔬2r+1, -1, if  𝔤= 𝔰𝔭2r, 1, if  𝔤= 𝔰𝔬2r, δ= 0, if 𝔤 =𝔰𝔬2r+1, 1, if 𝔤 =𝔰𝔭2r, -1, if 𝔤 =𝔰𝔬2r. Let ZV(l) be the central elements in Uq𝔤 defined by ZV(l) = ε idqtrV z 12 l and let ZV(l) = l0 ZV(l) u-l and ZV- = l0 ZV(-l) u-l. Then π0 ZV+ u - z q-q-1 - u 2 u 2 -1 = z q-q-1 u+q u-q-1 u-εqδ u+1 u-1 u-εq-δ σρ i=1 r u-ε L i -2 q-1 u-ε L i 2 q-1 u-ε L i -2 q u-ε L i 2 q .

Proof.
As operators on L0V, z 12 = z q ε1 , ε1 + 2ρ - ε1 , ε1 + 2ρ + 0, 0+ 2ρ =z and z 12 l = zl. So, as operators on L0, ZV(l) =ε id qtrV z 12 l = zl ε dimq V and, since ε dimq V = z-z-1 q-q-1 +1 , ZV+ = l0 ε dimq V zl u-l = ε dimq V 1 1-z u-1 = z-z-1+ q-q-1 q-q-1 1-z u-1 . A similar calculation yields ZV+ u = z-z-1+ q-q-1 q-q-1 1-z u-1 and z-z-1 +q+q-1 q-q-1 1- z-1 u-1 (see [BB], the last displayed equation in the proof of Lemma 7.4). Amazingly, as operators on L0,
ZV+ - ZV(0) + z q-q-1 - 1 u 2 -1 = z q-q-1 1- z-1 u-1 1-z u-1 u+q u-q-1 u+1 u-1 6.21
and
ZV- - ZV(0) + -z-1 q-q-1 - 1 u 2 -1 = -z-1 q-q-1 1-z u-1 1- z-1 u-1 u-q u+q-1 u+1 u-1 . 6.22

By (5.39), Yk acts on L0 Vk as z 12. If Lμ is an irreducible Uq 𝔤 -module in L0 Vk-1 , then (5.23) and (5.26) give that Yk acts on the Lλ component of Lμ V by the constant ε q 2cλμ = ε q y+2cλ/μ , if μλ, ε q -y -2 cμ/λ , if μλ, = z q 2c λ/μ , if μλ, z-1 q -2cμ/λ , if μλ, where cλ/μ is computed in (6.14) (see [OR]). By induction, as in [OR, Theorem 6.3(b)], the irreducible Wk-module W k μ/0 = W k μ has a basis vT indexed by up-down tableaux T = T(0), T(1), , T(k) , where T(0)=, T(k)=μ, and T(i) is a partition obtained from T(i-1) by adding or removing a box, and Yi vT = z q 2c b , if  b= T(i) / T(i-1) , z-1 q -2cb , if  b= T(i-1) / T(i) . Thus i=1 k-1 u-Yi 2 u- q-2 Yi-1 u- q 2 Yi-1 u - Yi-1 2 u- q 2 Yi u- q-2 Yi acts on the Lμ W k-1 μ isotypic component in the Uq𝔤 -module decomposition

L0 Vk-1 μ Lμ W k-1 μ , 6.23
by
i=1 k-1 u-ε q 2 c T(i) , T(i-1) 2 u- q-2 q -2 c T(i), T(i-1) u-ε q2 q -2c T(i), T(i-1) u-ε q -2c T(i), T(i-1) 2 u-ε q2 q -2c T(i), T(i-1) u-ε q-2 q 2c T(i), T(i-1) 6.24
for any up-down tableaux T of length k and shape μ. If a box is added (or removed) at step i and then removed (or added) at step j, then the i and j factora of this product cancel. Therefore, (
6.24) is equal to
bμ u-z q 2cb 2 u-z-1 q -2cb+1 u-z-1 q -2cb-1 u-z-1 q -2cb 2 u-z q 2cb+1 u-z q 2cb-1 6.25
(see [
BB], the next to last displayed equation in the proof of Lemma 7.4). Simplifying one row at a time, bμ u-z-1 q -2cb-1 u-z-1 q -2cb+1 u-z-1 q -2cb u-z-1 q -2cb = i=1 r u-z-1 q-2-i u-z-1 q -2μi-i+1 u-z-1 q -2-i-1 u-z-1 q-2μi-i = u-z-1 q2r u-z-1 q20 i=1 r u-z-1 q -2μi-i+1 u-z-1 q -2μi-i if μ=μ1μr. It follows that (6.25) is equal to
u-z-1 q2r u-z-1 u-z u-z q-2r i=1 r u-z-1 q -2μi-i+1 u-z-1 q -2μi-i u-z q2μi-i u-z q2μi-i+1 = u-ε qδ u-z-1 u-z u-ε qδ evμ+ρ i=1 r u-ε L i -2 q-1 u-ε L i 2 q-1 u-ε L i -2 q u-ε L i 2 q 6.26
where δ= 0, if 𝔤 =𝔰𝔬2r+1, 1, if 𝔤 =𝔰𝔭2r, -1, if 𝔤 =𝔰𝔬2r, since evμ+ρ L i 2 = evμ+ρ K 2 ε i = q μ+ρ , 2 ε i = q 2 μi + y-2i+1 = q y+1+2 μi-i = εz q 2 μi -i +1 and z-1 q2r = ε q-2r qδ q2r = ε qδ. Combining (
6.21) and (6.26), the identity (4.15) gives that Zk+ - Zk(0) + z q-q-1 - 1 u2-1 = Z1+ - Z1(0) + z q-q-1 - 1 u2-1 i=1 k-1 u-Yi 2 u-q-2 Yi-1 u-q2 Yi-1 u- Yi-1 2 u-q2 Yi u-q-2 Yi , acts on the Lμ W k-1 μ isotypic component in the Uq𝔤Wk-1-module decomposition in (6.23) by z q-q-1 1-z-1u-1 1-zu-1 u+q u-q-1 u+1 u-1 u-ε qδ u-z-1 u-z u-ε qδ evμ+ρ i=1 r u-ε L i -2 q-1 u-ε L i 2 q-1 u-ε L i -2 q u-ε L i 2 q = z q-q-1 u+q u-q-1 u+1 u-1 u-εqδ u-εq-δ evμ+ρ i=1 r u-ε L i -2 q-1 u-ε L i 2 q-1 u-ε L i -2 q u-ε L i 2 q .

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