Ngo Van Djnh Tap chi KHOA HOC & CONG NGHE 118(04); 203 - 207
O N T H E S C H U R M O D U L E S O F G L ^ ( C ) (Vg modun Schur cua nhdm GLm{C))
N g o V a n Dinh^
A b s t r a c t . This paper is a short review of the construction of Schur modules for the general linear group GLm{C), in which we make clear some points and complete some elementary proofs
Tom t d t . Muc dich ciia bai b i o nay la trinh bay cau triic cua cac modun Schur £ * cua nhom tuygn tinh tdng quAt GLm(C) = GL{E), trong do E la mot khong gian vectd phu'c m, chigu Cac modun Schur E^ la cac bilu diSn bat kha qui da thiic ciia nh6in GI.n,(C) dUdc tham s6 hoa bSi cac bang Young A v6i nhilu nhat m hang. Cac modun nay dUdc dinh nghia nhir vat the pha dung cua ho cac anh xa da tuygn tinh ip : £ " * -> F, tflr ttch Dl-cic E"^ den cac C-modun F, thoa man ba tfnh chit dac trUng (cac tinh chat (1), (2), (3) trong phan gidi thieu du6i). Trong mgnh dg 1, chiing toi chiJng minh rfing ta co the thay thg tinh ch^t d&c trung (3) bdi mpt dieu kien ddn gian hdn MQi d6iig cSu ip : E ^ F gifla cac C-mfldun hiiu han sinh deu cam sinh mot d6ng cSu 'p'' : E^ -^ F^. Chiing toi chiing minh trong menh dg 2 rSng ngu ip la mpt ddn cau thi tp ciing la ddn can va ngUOc lai Cufli ciing, chiing toi chiing minh trong menh de 3 cong thiic cij thi ciia t i c dpng cua dai s6 Endc(£) len B*.
Keywords: Schur module. Young diagram. Young tableau, complex representation, general linear group.
INTRODUCTION
We are interested in the problem to describe finite-dimensional irreducible representa- tions of the general linear group GLni(C) = GL{E), where £^ is a m—dimensional complex vector space, and to decompose finite-dimensional representations into irreducible compo- nents. For more details of these, we refer the readers to [1], [2], or [3].
The aim of this paper is to give elementary proofs for some properties of Schur modules E^ which are irreducible polynomial representations of GL^(C), parametrized by Young diagrams A with at most m rows.
Let US recall that a Young diagram is a collection of boxes arranged in left-justified rows, with a (weakly) decreasing number of boxes in each row. Listing the number of boxes in each row we obtain a partition of the integer n that is the total number of boxes.
Conversely, every partition of n corresponds to a Young diagram. We usually identify a partition, denoted by A, with the corresponding ''•agram. It is given by a sequence of weakly decreasing positive integers written A = (Ai, A2, - •, A^,). One writes by A ^ (d^'.rfj^,... ,d°») for the partition that has a^ copies of the integer di,l < i < s. The notation A h n is used to say that A is a partition of n, and |A| is used for the number partitioned by A. Any way of putting a positive integer per box of a Young diagram will be called a numbering or filling of the diagram. A Young tableau, or simply tableau, is a filling that is:
L weakly increasing across each row;
'Giing vien To&n, trirdng D?i hoc Khoa hoc, Dai hoc Thai Nguygn Email dinh.ngoStnus edu v
Ngo Van Dinb Tap chi KHOA HOC & CONG NGHE 118(04): 203 - 207 2. strictly increasing down each column.
We say that such a tableau is a tableau on the diagram A, or that A is the shape of the tableau. A standard tableau is a tableau in which the entries are the numbers in [n], set of positive integers from 1 to n, each occurring once.
Throughout this paper, £ is a complex vector space of dimension m. We will write the Cartesian product of n copies of E in the form
E'''' ^ExEx •xE.
We denote by E^^ the cartesian product of n = |A] copies of E, but labelled by the n boxes of A, So an element v of E"^^ is given by specifying an element of B for each box in A.
Consider maps ip : E^^ -i- F from E^^ to a C - m o d u l e F, satifying the following three properties:
(1) ip is C—multihnear.
(2) tp is alternating in the entries of any column of A.
(3) For any v in E^^,ip(v) = X^'^(tu), where the sum is over all w obtained from v by an exchange between two given columns, with a given subset of boxes in the right chosen column.
Definition 1 ([!])• The universal target module for such maps (p is called Schur m,odule and denoted by E^.
By definition, E^ is a complex vector space and we have a map E^^ —» E^, that we denote v >-*• u^, satisfying (1) — (3), and such that for any <fi : E''^'^ —¥ F satisfying (1) — (3), there is a unique homomorphism ip : E^ -^ F oi C—modules such that <fi(v) = ip(v^) forall v in E^'^. In fact, the relation (3) in the definition of Schur module, one allows to interchange only the entries between two adjacent columns. This is stated by the proposition 1.
We have (see in [1])
S^ = A^'E ®c • •' ®c A>''E/Q^(E),
where A is numbered down the columns from left to right, pi is length of the i"" colunm of A and Q^iE) is the submodule generated by all elements of the form Av — ^ Aw, the sum is taken over all w obtained from v by the exchange procedure described in (3). It follows directly from the definition that the construction of E'^ is functorial in E. This means that any homomorphism ip : E —¥ F determines a homomorphism f^ -. E^ —^ F^ such that the following diagram is commutative
XX ^
(')
E^
It is easy to check that if 93 is a surjection of C—modules then ip^ is also surjective. A similar result for injective maps is not true in general. However it is true in case ^p is &, monomorphism. We give an elementary proof for this case in the proposition 2.
Ngo Van Dinh Tap chi KHOA HOC & CONG NGHE 118(04): 203 - 207- Let { e i , e 2 , - . . ,em} is an order set of elements of £ ; T is a munbering of shape A with elements in [m]. By replacing any z in a box of T by the element e, we have an element of E^^. The image of this element in E^ denoted by ey. By the functoriality of the construction of E^, any endomorphism of E determmes an endomorphism of E^, this gives a left action of algebra Endc(S) on E^,v^ i-^ g .v^ such that the following diagram is commutative
-E-'i,*)
£>•- -E\
Proposition 3 give the exact formula for this.action. In particular, the group GL(E) of automorphisms of E acts on the left on E'^. Given a basis of E, one can identify E with K"^, then Endc(£^) = MmC is the algebra of m x m matrices. Therefore M ^ C acts on E^, as does the subgroup GLm(C). So Schur module E^ is a finite dimention polynomial representation of GLm(C). We refer the reader to [1] for the proof of irreducibility of E^.
A c k n o w l e d g e m e n t s . The author gratefully acknowledges the many helpful suggestions of Prof. Dr DSc Do Ngoc Diep during the preparation of the paper.
MAIN RESULTS
Proposition 1. One obtains the same module E^ if, m relation (3), one allows to inter- change between two adjacent columns only.
Proof Assume that for any v in E^^,(p{v) = I ] v ( ^ ) . where the sum is over all elements w in E^^ obtained from u by an exchange between two given adjacent columns, with a given subset of boxes in the right chosen column. We will prove that relation (3) is also satisfied, this means that the formula, ip(v) = Yl'Pi'^)^ holds when w obtained from v by an exchange between two given columns, with a given subset of boxes in the right chosen column. It is sufficient to prove for the exchange top k boxes in the (/ -|- 2)"'' column with /''' column of u, (1 < / < Ai - 2 ) .
Now let a i , a 2 , . . . ,ti,i, be the entries of the /"' column, respectively from the top to the bottom; yi,y2,--- ,ytii+i be the entries ofthe (/-hl)^' column, respectively from the top to the bottom; and xi,X2,-• • ,x^,_^_2 be the entries ofthe ( / + 2)"'^ column, respectively from the top to the bottom, where fc < pi+2 < W-i-i ^ t^i- This means that
O l
" 2
"/.,
S l
!/2
!/ii,+i I I 1 2
Ngo Van Dinh Tep chi KHOA HOC & CONG NGHE I I 8(04): 203-207 We have
(ji-- j k ) O l 112
fe
Xl
Xk
%.
Vh
Xk+l
)
CJ'I,- •.3k){il,- ,ik) XlXk O i ,
a j t Vh
Vit
Xk+1
)
= E E ^
(!l,-.2fc)(jlr-Jfc) Xl
Xk a,.
a„
fc.
Vil.
Xk+l
- E ^
( • 1 . • ,i»)
(
I I
Xk
! / l
!/2
S p H .
".,
I t + 1
where ( j i , . . . , jfc) and ( i i , . . . , 4 } are ^ - t u p l e s satisfied
lijl < J2< ••• < jki w+i and 1 ^ il < 22 < • • • < j * !* w . D Proposition 2. Let ip : E ^ F be a homomorphism of finitely generated free C—modules.
Then the following statements are equivalent (i) if is a monomorphism;
(ii) <p^ : E^ —> F^ is a monomorphism for all A;
(iii) ip^ IS a monomorphism for some A with at most m rows, m = rank{E).
Proof, [i) =^ {ii). Consider v^ € kenp^, we liave
q{^^\v)) = p\v>-) = 0€F^^ vj'^W e Q\F).
Since ^p^^ is monomorphism then v e Q^(E), i.e., u'^ is zero of f''^.
(ii) ^ (iii). This is obvious.
(iii) ^ (j). Let {ei,e2, • •. ,em} be a basis of E. We only need to show that
'piei),ip(e2),... ,ip(em) are linearly independent in F. Suppose that these elements are linearly dependent in F^ i.e., we have a hnear combination ^ ocjip(ej) = 0, where there
j = i
exists at least a coefficient Oj is not zero. Let jo be the largest number such that QJQ ^ 0.
Let's consider element v in E^^ defined by:
Ngo Van Dinh Tap chi KHOA HOC & CONG NGHE 1 i 8(04)- 203 - 207
+ ) If fc > Jo then V has entry at the first box of the JQ^ row is JZ oij^j, and entries at the other boxes are e^ if its box in the z"^ row.
+ ) If fc < jo then V has entry at the first box ofthe bottom row is ^ Oj-ej, entries at different boxes of this row are ej^, and entries at the other boxes are e^ if its box in the i"* row.
For a such element v then q(^^^(v)) is zero in F^^. In the other hand, it is clear that the statement v^ ^ 0 follows (p^(v^) is not zero which contradicts the assumptions. D Proposition 3. Assume that g = (gij) is an element in MmC. IfT has entries
j\t32i- • • >jn it^ its n boxes (ordered arbitrary), then P . er = X ] 5^iJi • • • 5'"J" •^^''
where the sum is taken over the rri^ fillings T' obtained from T by replacing the entries {3ij2,---,jn) by (ii,i2,---,in), from [mj.
Proof Suppose that ( e i , . . . , e^} is a basis of E. We denote by e{T), the element in E^^
which has image er in E^. We have gsj = J2 9ij^f Then
1 = 1
g''\e(T)) ^Y.9H3I • • • StnjMT') ^ 9 -er = Y^9rur • • 9^.jneT'-
By the diagram (**), the proof is complete. • REFERENCES
II] W. Fulton, Young Tableaux - With Aphcations to Representation Theory and Geometry, Cambridge University Press, 1997.
[2] W. Pulton, J. Harris, Representation Theory - A first course, Graduate texts in mathematics, Springer- Veriag, New York, USA, 1991.
[3] A A. Kirillov, Elements ofthe Theory of Representations, Springer-Verlag, New York, USA, 1976.
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