Derangements and Tensor Powers of Adjoint Modules for sln
GEORGIA BENKART∗ [email protected]
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA
STEPHEN DOTY [email protected]
Department of Mathematical and Computer Sciences, Loyola University Chicago, Chicago, Illinois 60626, USA Received July 18, 2001; Revised January 17, 2002
Abstract. We obtain the decomposition of the tensor spacesl⊗nkas a module forsln, find an explicit formula for the multiplicities of its irreducible summands, and (whenn≥2k) describe the centralizer algebraC=Endsln(sl⊗nk) and its representations. The multiplicities of the irreducible summands are derangement numbers in several important instances, and the dimension ofCis given by the number of derangements of a set of 2kelements.
Keywords: derangements, centralizer algebras, walled Brauer algebras, tensor powers, adjoint representation
Introduction
Weyl’s celebrated theorem on complete reducibility says that a finite-dimensional module X for a finite-dimensional simple complex Lie algebra gis a direct sum of irreducible g-modules. However, to determine an explicit expression for the multiplicities of the irre- ducibleg-summands ofXoften is a very challenging task. In this note we assumeg=sln, the simple Lie algebra ofn×n matrices of trace 0 overC, and viewsln as ag-module under the adjoint action x·y = [x,y]. We take X to be thek-fold tensor power ofsln. Using combinatorial methods and results developed in [2], we establish an explicit descrip- tion of the irreducibleg-summands ofsl⊗nk (Theorem 1.16) and determine an expression for their multiplicities (Theorem 2.2). As a consequence of our formula, we obtain the following results, expressed in terms of the number Dk of derangements of{1, . . . ,k}:
For n ≥ 2k, the dimension of the space of g-invariants in sl⊗nk is Dk; the multiplic- ity of sln insl⊗nk is Dk+1; and the dimension of the centralizer algebraC = Endg(sl⊗nk) isD2k.
In Section 3, we identify the centralizer algebra C with a certain subalgebra of the walled Brauer algebra Bk,k(n). This subalgebra has a basis indexed by derangements of {1, . . . ,2k}. We then give a description (forn ≥2k) of the irreducible modules forC, and obtain the “double centralizer” decomposition of the tensor spacesl⊗nk as a bimodule for C×g.
∗Supported in part by NSF Grant no. 9970119.
1. The tensor product realization
The general linear Lie algebragln =sln ⊕CI of alln×ncomplex matrices acts onsln
via the adjoint action, and the identity matrix I acts trivially. Hence, there is no harm in assuming thatgisglnrather thanslnacting onsl⊗nkin what follows; the results are exactly the same. This enables us to label the irreducible summands by pairs of partitions and to apply known results on the decomposition of tensor products forgln.
Lethdenote the Cartan subalgebra ofg=glnof diagonal matrices, and leti :h→Cbe the projection of a diagonal matrix onto its (i,i)-entry. The irreducible finite-dimensional g-modules are labeled by their highest weight, which is an integral linear combination n
i=1κii withκ1≥κ2 ≥ · · · ≥κn. By lettingλ=(λ1 ≥λ2 ≥ · · ·) denote the sequence of positiveκi andµ=(µ1≥µ2≥ · · ·) be the partition determined by the negativeκi, we may associate to each highest weight a pair of partitions (λ, µ). For example, forg=gl12 the highest weight
31+22+23+24+5−410−511−512
is identified with the pair of partitionsλ =(3,2,2,2,1) 10 andµ = (5,5,4) 14.
Therefore, the set of highest weights forg-modules is in bijection with the set of pairs of partitions such that the total number of nonzero parts does not exceedn.
Let V = Cn be the natural representation of g = gln on n ×1 matrices by ma- trix multiplication. The dual module V∗ may be identified with 1×n matrices, where the g-action is by right multiplication by the negative of an element x ∈ g. The matrix product
V⊗V∗ →gln =sln⊕CI, u⊗w∗→uw∗ (1.1)
is ag-module isomorphism which allows us to identifyglnwithV ⊗V∗.
Let{v1, . . . , vn}denote the standard basis ofV, wherevi is the matrix having 1 in the ith row and 0 everywhere else. Assume{v∗1, . . . , vn∗}is the dual basis inV∗, so thatv∗i has 1 in itsith column and 0 elsewhere. Thecontraction mapping c : V⊗V∗ →V ⊗V∗is defined using the trace by
c(u⊗w∗)=tr(uw∗) n
=1
v⊗v∗. (1.2)
Under the isomorphism in (1.1),v⊗v∗is mapped to the matrix unitE,∈gln. Therefore, we may identify the image ofcwithCI, and the kernel ofcwithsln.
Asc2=nc, the mappingp=(1/n)cis an idempotent. It is the projection onto the trivial summandCI, and id−pis the projection ontosln. These idempotents are orthogonal,
p(id−p)=0=(id−p)p,
and satisfy id=(id−p)+p. (Here id is the identity map onV⊗V∗.)
In order to identifysl⊗nkwith a summand of
M =V⊗k⊗(V∗)⊗k∼=(V ⊗V∗)⊗k∼=gl⊗nk, (1.3) we define the contraction mapci,j to be the contractioncapplied to theith factor ofV⊗k and the jth factor of (V∗)⊗kaccording to
ci,j(u1⊗ · · · ⊗uk⊗w∗1⊗ · · · ⊗w∗k)
= tr(uiw∗j) n
=1
u1⊗ · · ·v⊗ · · · ⊗vk⊗w1∗⊗ · · · ⊗v∗· · · ⊗wk∗,
where v is placed in the ith slot of V⊗k andv∗ in the jth slot of (V∗)⊗k. As before, c2i,j =nci,j, so that
pi = 1
nci,i (1.4)
is an idempotent.
Proposition 1.5 kerp1∩kerp2∩ · · · ∩kerpk =(id−p1)(id−p2)· · ·(id−pk)M.
Proof: The idempotents pi commute and satisfy pi(id− pi) = 0. For J a subset of {1, . . . ,k}, let pJ =
j∈J pj. Setqj =id−pjandqJ =
j∈Jqj. Then
M =
J⊆{1,...,k}
pJcqJM,
whereJc= {1, . . . ,k}\J. This can be argued by induction onk. Note that the sum is direct because for any fixed choice of subset J, the idempotent pJcqJ acts as the identity on pJcqJM and annihilates the remaining terms pJcqJM withJ = J. Whenever j ∈ Jc, thenpJcqJMis not contained in kerpj. Therefore, from the decomposition ofMabove, it is easy to see that kerp1∩kerp2∩ · · · ∩kerpk=(id−p1)(id−p2). . .(id−pk)M.
Henceforth, let
e=(id−p1)(id−p2). . .(id−pk) (1.6)
so that
eM ∼=sl⊗nk. (1.7)
The centralizer algebra Endg(M) of transformations commuting with the action ofg=gln
onM =V⊗k⊗(V∗)⊗kwas investigated in [2], where it was shown to be a homomorphic image of a certain algebra Bk,k(n) of diagrams with walls. A diagram in Bk,k(n) consists
of two rows of vertices with 2kvertices in each row. There is a wall separating the firstk vertices on the left in each row from thekvertices on the right. Each vertex is connected to precisely one edge but with the requirement that horizontal edges must cross the wall, but vertical edges cannot cross. The productd1d2of two diagramsd1andd2is obtained by placingd1aboved2, identifying the bottom row ofd1with the top row ofd2, and following the resulting paths. Cycles in the middle are deleted, but there is a scalar factor, which is n to the number of middle cycles. For example, in B5,5(n) we would have the following product,
The groupSk×Sk acts onM, where the first copy of the symmetric groupSk acts on the firstkfactors and the second copy on the nextkfactors by place permutation. These actions commute with the g-action, and so afford transformations in Endg(M). There is a representationφ : Bk,k(n) → Endg(M) of the algebra Bk,k(n) onM which commutes with theg-action. Under this representation, the diagrams inBk,k(n) having no horizontal edges are mapped to the place permutations coming from Sk×Sk. The identity element inBk,k(n) is just the diagram with each node in the top row connected to the one directly below it in the second row, and it maps to the identity transformation in Endg(M). Underφ, a diagram such as the one pictured below is mapped to a contraction mapping (in this case toc3,1).
(1.8)
It is shown in [2] that the algebra Endg(M) is generated bySk×Skand the contraction mapsci,j, and the above mappingφis an isomorphism ifn ≥2k. Moreover [2] describes the projection maps onto the irreducible summands ofMin the following way.
Suppose for some integer r satisfying 0 ≤ r ≤ k that s = {s1, . . . ,sk−r}and t = {t1, . . . ,tk−r}are ordered subsets of{1, . . . ,k}of cardinalityk−r, and define the following product
cs,t def=cs1,t1· · ·csk−r,tk−r (1.9)
of the contraction mapscsi,ti. Thencs,t belongs to the centralizer algebra Endg(M). There is a corresponding product of diagrams in Bk,k(n) like the one displayed in (1.8), whichφ maps ontocs,t.
Assumeλ = (λ1 ≥ λ2 ≥ · · ·) is a partition ofr. Associated toλis its Young frame or Ferrers diagram having λi boxes in theith row. A standard tableau is a filling of the boxes in the diagram ofλin such a way that the entries increase from left to right across each row and down each column. LetT be a standard tableau of shapeλwith entries in sc= {1, . . . ,k}\{s1, . . . ,sk−r}. Associated toT is its Young symmetrizer
yT =
ρ∈RT
ρ
γ∈CT
sgn(γ)γ
, (1.10)
where the first sum ranges over the row group ofT, which consists of all permutations in Skthat transform each entry ofT to an entry in the same row, and the second sum is over the column group ofT of permutations that move each entry ofT to an entry in the same column. For example,
y =(id+(1 5))(id−(1 4)),
which belongs to the group algebraCSk of the symmetric groupSk. The map yT is an essential idempotent, that is, there is an integermso that yT2 =myT.
Similarly, assume for some partitionµrthatT∗is a standard tableau of shapeµwith entries chosen fromtc= {1, . . . ,k}\{t1, . . . ,tk−r}. The mapping
yTyT∗cs,t (1.11)
is an essential idempotent in Endg(M). (Note that here we are supposing that yT acts on the factors in V⊗k and yT∗ on the factors in (V∗)⊗k by place permutations, and that id is the identity map onV⊗kor (V∗)⊗k, respectively.) Moreover,yTyT∗cs,tM is isomorphic to the irreducibleg-moduleL(λ, µ) having highest weight given by the partitionsλandµ.
The collection of all mapsyTyT∗cs,tasr=0,1, . . . ,k;s,trange over all possible choices of ordered subsets of cardinalityk−rin{1, . . . ,k};λandµrange over all partitions ofr;
andT (resp.T∗) ranges over all standard tableaux of shapeλ(resp.µ) with entries insc (resp. intc), give all the projections onto the irreducible summands ofM(this can be found in [2]).
Now for the idempotentein (1.6) we may apply the standard result, Endg
sl⊗nk ∼=Endg(eM)=eEndg(M)e|eM, (1.12)
(see for example, [4, Lemma 26.7] or [1, Proposition 1.1]).
Lemma 1.13 Assume y=yTyT∗cs,t. If cs,tcontains one of the contraction maps cj,j for some j =1, . . . ,k,then ey=0=ye.
Proof: The mappingsyT,yT∗,csi,ti,i =1, . . . ,k−r, all commute with one another as they operate on different tensor factors. If one of the contraction maps inyequalscj,j=npj, then moving it to the far right produces a productpje= pj(id−pj)
=j(id−p)=0 in ye, soye=0. The argument foreyis similar.
In [2, Definition 2.4] (compare also [6]) a certain simple tensorxT,T∗,s,t =u1⊗ · · · ⊗ uk⊗w1∗⊗ · · ·wk∗ofMis constructed via the algorithm
up =
v1 if p∈s
vj if p∈scandpis in the jth row ofT
(1.14) w∗p =
v∗1 if p∈t
v∗n−j+1 if p∈tcandpis in the jth row ofT∗
Wheny=yTyT∗cs,tis applied to the simple tensorx =xT,T∗,s,tthe resultyxis a nonzero highest weight vector fory M. Moreover, all the highest weight vectors inM are produced in this fashion.
Observe that the factors inxlie in{v1, . . . , vr, v1∗, v∗n, . . . , v∗n+1−r}. When the pair (si,ti) belongs to (s,t), then the vectorv1 lies in slotsi inV⊗k, andv1∗lies in slotti in (V∗)⊗k. Replace v1 byvr+i andv1∗ byvr∗+i in slotssi andti for i = 1, . . . ,k−r, to produce a new simple tensor x. Then yx = yx, as the effect of applying a contraction tov1⊗ v∗1 or to vr+i ⊗vr∗+i is the same. However, if si = ti for anyi = 1, . . . ,k −r, then pjx = 0 for all j. The reason for this is that the vector factors of x form a subset of {v1, . . . , vr, vr+1, . . . , vk, vn∗, . . . , v∗n+1−r, vk∗, . . . , vr∗+1}. Ifn ≥ 2k, these are all distinct.
Assi =tifor anyi =1, . . . ,k−r, slot jon the left and slot jon the right do not contain a pair of dual vectors (of the formv, v∗). Therefore pjx=0 for all j andex=x.
These calculations show that yeM = 0, as it contains yex = yx = yx, which is a maximal vector of weight (λ, µ). But thenyeM ⊆y M, and the irreducibility ofy Mforces yeM=y M. To summarize we have
Proposition 1.15 Assume n ≥ 2k. If y = yTyT∗cs,t and si =ti for any pair(si,ti)in (s,t),then yeM= y M,an irreducibleg-module of highest weight(λ, µ),whereλis the shape of T andµis the shape of T∗.
Letcjdenote the diagram inBk,k(n) corresponding to the contractioncj,j, but scaled by a factor of 1/n. Then under the representationφ: Bk,k(n)→ Endg(M),cj is sent to pj, andb=k
j=1(1−cj) is mapped to the idempotente.
Let us consider the subspace Aspanned by the diagramsd having no forbidden pairs.
By a forbidden pair we mean that theith node on the left is connected to theith node on the right of the wall either in the top or in the bottom row ofdfor somei =1, . . . ,k.
We claim that the mapBk,k(n)→b Bk,k(n)bis injective on the subspaceAof diagrams with no forbidden pairs. Indeed,
d∈Aadd →
d∈Aadbdb =
d∈Aadd + f, where ad ∈Cand f is a linear combination of diagrams inBk,k(n) having at least one forbidden pair. The reason for this is that when diagrams are multiplied, the horizontal edges in the top
row of the top diagram and the horizontal edges in the bottom row of the bottom diagram always appear in the resulting product diagram.
Assume y = yTyT∗cs,t is such thatsi =ti for anyi =1, . . . ,k−r, and consider the mapyM =yeM → eyeMgiven by yem → eyem. We have argued thatyeM =yM, an irreducibleg-module. Therefore, this map either is an injection or is identically zero. In the latter case,eyemust be the zero transformation in Endg(M). But there is a linear combination zof diagrams inAwhich maps toyunder the representationφ, andbzb→eye. Because the productbzbis nonzero, and the representationφis faithful onM ifn ≥2k,eyemust be nonzero. ThereforeyM =yeM→eyeMis an injection. But it is clearly surjective and ag-module map, soeyeM∼=yM, an irreducibleg-module with highest weight (λ, µ). We have proved part (1) of the following:
Theorem 1.16 Assume n≥2k,g=gln,and M=V⊗k⊗(V∗)⊗k.
(1) Let y= yTyT∗cs,t,where si =ti for any i =1, . . . ,k−r . Then eye(eM)=eyeM is an irreducibleg-submodule of eM of highest weight(λ, µ)whereλis the shape of T andµis the shape of T∗.
(2) sl⊗nk ∼=eM =
yeyeM,where the sum is over all y=yTyT∗cs,tsuch that si=ti for any i .
Proof: What remains to be shown is thateM =
yeyeM. Observe that because
M =
T,T∗,s,t
yTyT∗cs,tM, (1.17)
([2, Theorem 2.11], compare also [7]),
eM=
y
eyM=
y
eyeM, (1.18)
where the sum is over ally =yTyT∗cs,t such thatsi =ti for anyi. We need to argue that the decompositioneM =
y(eye)eM (over suchy) is direct.
We have shown previously thatyM = yeMand the map E : yeM → eyeM given by yem →eyemis an isomorphism ofg-modules fory=yTyT∗cs,tsuch thatsi =ti for any i. Fix one such idempotentyand consider the intersection
eyeM∩
y=y
eyeM
ofeyeMwith the sum over the remaining ones. Then eyeM∩
y=y
eyeM E→−1 yeM∩
y=y
yeM=yM ∩
y=y
yM
ButyM∩
y=yyM=0 by (1.17). Thus, the sum in (1.18) is direct and we have (2).
2. Multiplicities
Knowing that
sl⊗nk∼=eM =
y
eyeM,
where the sum is over all y = yTyT∗cs,t such thatsi = ti for anyi, we may deduce the multiplicity of a particular irreducible summand insl⊗nklabelled by (λ, µ), whereλ, µr andr =0,1, . . . ,k. That multiplicity is the number ofy=yTyT∗cs,twithT having shape λ, T∗having shapeµ, andcs,thaving no pairssi=ti. Counting the number ofcs,twith at least j factors of the formc,, we have (kj) for the choice of those contractions, (k−k−r−j j) choices for the remaining si’s in s, and (k−k−r−j j) for the rest of the ti’s in t, and (k−r− j)! for the number of ways to pair the chosensi’s with the chosenti’s. Thus, the number of suchyTyT∗cs,twith at least jcontractions of the formc,is
k j
k−j k−r−j
2
(k−r−j)!fλfµ = k
j
k−j r
2
(k−r−j)!fλfµ, (2.1)
where fλ(resp. fµ) is the number of standard tableaux of shapeλ, (resp.µ). Therefore, by the inclusion-exclusion principle, we have the following result.
Theorem 2.2 When n ≥ 2k,the multiplicity mkλ,µ insl⊗nk of the irreducibleg = gln- module L(λ, µ)with highest weight(λ, µ),whereλ, µr,is
mkλ,µ= fλfµ k−r
j=0
(−1)j k
j
k− j r
2
(k−r−j)!
. (2.3)
For a partitionλofr, the number fλ of standard tableaux of shapeλis given by the well-known hook length formula
fλ= r!
h(λ), whereh(λ)=
(i,j)∈λhi,j,the product of thehook lengthsof the boxes ofλ. Thus,hi,jis the number of boxes in the (i,j) hook ofλ: the number of boxes to the right of (i,j) plus the number of boxes below (i,j) plus 1.
As a result, the expression for the multiplicity of the summand labelled by (λ, µ) also can be written as
mkλ,µ= 1 h(λ)h(µ)
k−r
j=0
(−1)j k!(k−j)!
j!(k−r−j)!. (2.4)
Let us consider a few interesting special cases. The multiplicity of the trivialg-module insl⊗nk(that is, the dimension of the space ofg-invariants) is
mk∅,∅= k
j=0
(−1)j k
j
(k− j)!=k!
k j=0
(−1)j 1
j! =Dk, (2.5)
which is the number of derangements on the set{1, . . . ,k}(permutations with no fixed elements). For small values ofk, this number is given by
k 1 2 3 4 5 6 7 8
Dk 0 1 2 9 44 265 1854 14,833 (2.6)
Next, we compute the number of times the adjoint modulesln =L( , ) occurs insl⊗nk. Using the fact thatslnis self-dual as ag-module, we see that the number of timesslnappears insl⊗nkis the number of times the trivial module appears insl⊗nk⊗sln =sl⊗(kn +1). Hence, the number of timesslnappears insl⊗nkis
mk, =Dk+1. (2.7)
This can also be derived from (2.4) which gives mk, =
k−1
j=0
(−1)jk!(k− j) j! =
k j=0
(−1)jk!(k−j) j!
=k k
j=0
(−1)jk!
j!+ k
j=1
(−1)j−1 k!
(j−1)! (2.8)
=k k
j=0
(−1)jk!
j!+k
k−1
j=0
(−1)j(k−1)!
j!
=k(Dk+Dk−1)=Dk+1.
The last equality in (2.8) is a linear recurrence relation satisfied by the derangement numbers (see for example, [3, (6.5)]).
For anyg-moduleX, X⊗X∗ ∼=End(X)
where the action on the right is (g·ψ)(x)=gψ(x)−ψ(gx) for allg ∈g,ψ∈ End(X), andx∈ X. Considering theg-invariants on both sides, we see that
(X⊗X∗)g∼=End(X)g=Endg(X). (2.9)
Now applying this to X=sl⊗nk ∼=X∗, we have Endg
sl⊗nk ∼=
sl⊗2kn g (2.10)
Consequently, dim Endg
sl⊗nk =m2k∅,∅=D2k, (2.11)
the number of derangements on a set of 2kelements.
We conclude by displaying the multiplicities mkλ,µ for k = 4. By double centralizer theory, it follows that
dim Endg
sl⊗nk =
λ,µr≤k
mkλ,µ 2.
The reader can verify that the squares of the numbers in the following tables do indeed sum toD8=14,833.
Example m4λ,µ:
3. The centralizer algebra
Now we consider the centralizer algebraC=Endg(sl⊗nk)=Endsln(sl⊗nk) and its represen- tation theory. As has already been pointed out in (1.12), we have an isomorphism
C∼=eEndg(M)e (3.1)
whereeis the idempotent defined in (1.6). We also have a representationφ : Bk,k(n)→ End(M) which commutes with theg-action onM. Thus the image of this representation lies in the commuting algebra Endg(M). In [2, Theorem 5.8] it was shown thatφinduces an algebra isomorphism
Bk,k(n)∼=Endg(M) (3.2)
forn≥2k. Letb∈ Bk,k(n) be given as above,b=k
j=1(1−cj) wherecjis the diagram corresponding to the contraction mapcj,j but scaled by 1/n. Thenφmapsb ontoe, and we obtain the following.
Proposition 3.3 Let n ≥ 2k. The map φ induces an algebra isomorphism between b Bk,k(n)b and C = Endg(sl⊗nk). Moreover, the set of all elements of the form bdb, as d ranges over all diagrams with no forbidden pairs,is a basis for b Bk,k(n)b.
Proof: The first claim follows from the remarks above, so only the second claim remains to be proved. We observe that left (resp., right) multiplication bybkills any diagram with a forbidden pair in its top (resp., bottom) row. Since the diagrams form a basis forBk,k(n), the result follows.
The basis statement of Proposition 3.3 provides another proof of (2.11), that the dimension of the centralizer algebraCisD2k. Indeed, the diagrams with no forbidden pairs are easily seen to be in bijective correspondence with the permutationsσon the set{1, . . . ,2k}such thatσ(i)=ifor alli =1, . . . ,2k. This correspondence is given by performing two “flips”, which take a walled Brauer diagram to the diagram obtained by first interchanging the rightmostkdots in its top and bottom rows and then switching corresponding dots on the two sides of the wall on the top row while retaining the edges.
Letr ≤ kand letλ, µbe fixed partitions ofr. In [2] Mλ,µwas defined to be the space spanned by all maximal vectors yx, where y = yTyT∗cs,t andx = xT,T∗,s,t (notation of (1.14)), for all pairss= {s1, . . . ,sk−r},t = {t1, . . . ,tk−r}of ordered subsets of{1, . . . ,k}, and all standard tableaux T (resp., T∗) of shapeλ (resp.,µ) with entries fromsc (resp., tc). Moreover, for n ≥ 2k, the Mλ,µ provide a complete set of pairwise nonisomorphic irreducible modules for the algebra Endg(M) (and hence also forBk,k(n)).
Lemma 3.4 Assume n ≥ 2k and let y = yTyT∗cs,t,x = xT,T∗,s,t. Then ey = 0if and only if si=ti for all pairs(si,ti)in(s,t). Hence eMλ,µ=0precisely when this condition can be satisfied,and in that case,eMλ,µis the linear span of all the nonzero eyx,y and x as above.
Proof: This follows from results in [2], Lemma 1.13, and its converse, which is in the paragraph before Theorem 1.16.
We remark that fory,xas in the preceding lemma, we haveeyx=eyx=eyex, where xis the modified simple tensor described in Section 1.
Moreover, it is easy to see thateMλ,µ=0 whenλ=µ= ∅andk=1, for in that case it is impossible to construct a y = yTyT∗cs,t satisfying the conditionsi =ti for all pairs (si,ti) in (s,t). In all other caseseMλ,µ=0 whenn≥2k.
Theorem 3.5 Assume n≥2k. The collection of all nonzero eMλ,µforλ, µpartitions of r,r = 0,1, . . . ,k,forms a complete set of pairwise nonisomorphic irreducible modules for the algebraC∼=b Bk,k(n)b.
Proof: It is well-known that ifu is an idempotent in an algebra A, the functoru(−) (sometimes called the Schur functor; see [5, 6.2]) takingA-modules tou Au-modules is an exact covariant functor which maps an irreducible module to either an irreducible module or zero. In the particular case thatA=Bk,k(n) andu=b, this functor takes the irreducible moduleMλ,µtobMλ,µ=eMλ,µ.
Theorem 3.6 Assume n≥2k. Then as a bimodule forC×g,
sl⊗nk∼=eM ∼= k
r=0
λ,µr
eMλ,µ⊗L(λ, µ),
where the decomposition is into pairwise nonisomorphic irreducible modules forC×g.
Proof: This follows from the previous results and standard double-centralizer theory.
Forn ≥ 2k the dimension of the irreducibleC-moduleeMλ,µ is given by mkλ,µ (see Theorem 2.2).
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