On the schur modules of GLm(c) - Ngô Văn Định

ON THE SCHUR MODULES OF GLm(C) (Về môđun Schur của nhóm GLm(C)) Ngô Vaˇn Định1 Abstract. 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. Tóm tắt. Mục đích của bài báo này là trình bày cấu trúc của các môđun Schur Eλ của nhóm tuyến tính tổng quát GLm(C) = GL(E), trong đó E là một không gian véctơ phức m chiều. Các môđun Schur Eλ là các biểu diễn bất khả qui đa thức của nhóm GLm(C) được tham số hóa bởi các bảng Young λ với nhiều nhất m hàng. Các môđun này được định nghĩa như vật thể phổ dụng của họ các ánh xạ đa tuyến tính ϕ : E×λ → F , từ tích Đề-các E×λ đến các C-môđun F , thỏa mãn ba tính chất đặc trưng (các tính chất (1), (2), (3) trong phần giới thiệu dưới). Trong mệnh đề 1, chúng tôi chứng minh rằng ta có thể thay thế tính chất đặc trưng (3) bởi một điều kiện đơn giản hơn. Mỗi đồng cấu ϕ : E → F giữa các C-môđun hữu hạn sinh đều cảm sinh một đồng cấu ϕλ : Eλ → Fλ. Chúng tôi chứng minh trong mệnh đề 2 rằng nếu ϕ là một đơn cấu thì ϕλ cũng là đơn cấu và ngược lại. Cuối cùng, chúng tôi chứng minh trong mệnh đề 3 công thức cụ thể của tác động của đại số EndC(E) lên E λ. 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 GLm(C) = GL(E), where E 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 Eλ which are irreducible polynomial representations of GLm(C), parametrized by Young diagrams λ 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 λ, with the corresponding diagram. It is given by a sequence of weakly decreasing positive integers written λ = (λ1, λ2, . . . , λm). One writes by λ = (da11 , d a2 2 , . . . , d as s ) for the partition that has ai copies of the integer di, 1 ≤ i ≤ s. The notation λ ` n is used to say that λ is a partition of n, and |λ| is used for the number partitioned by λ. 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: 1. weakly increasing across each row; 1Giảng viên Toán, trường Đại học Khoa học, Đại học Thái Nguyên. Email: dinh.ngo@tnus.edu.vn 2. strictly increasing down each column. We say that such a tableau is a tableau on the diagram λ, or that λ 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, E is a complex vector space of dimension m. We will write the Cartesian product of n copies of E in the form E×n = E × E × · · · × E. We denote by E×λ the cartesian product of n = |λ| copies of E, but labelled by the n boxes of λ. So an element v of E×λ is given by specifying an element of E for each box in λ. Consider maps ϕ : E×λ → F from E×λ to a C−module F, satifying the following three properties: (1) ϕ is C−multilinear. (2) ϕ is alternating in the entries of any column of λ. (3) For any v in E×λ, ϕ(v) = ∑ ϕ(w), 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 ([1]). The universal target module for such maps ϕ is called Schur module and denoted by Eλ. By definition, Eλ is a complex vector space and we have a map E×λ → Eλ, that we denote v 7→ vλ, satisfying (1)− (3), and such that for any ϕ : E×λ → F satisfying (1)− (3), there is a unique homomorphism ϕ˜ : Eλ → F of C−modules such that ϕ(v) = ϕ˜(vλ) forall v in E×λ. 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]) Eλ = Λµ1E ⊗C · · · ⊗C ΛµlE/Qλ(E), where λ is numbered down the columns from left to right, µi is length of the ith column of λ and Qλ(E) is the submodule generated by all elements of the form ∧v−∑∧w, 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 Eλ is functorial in E. This means that any homomorphism ϕ : E → F determines a homomorphism ϕλ : Eλ → F λ such that the following diagram is commutative E×λ ϕ×λ // p  F×λ q  Eλ ϕλ // F λ. (∗) It is easy to check that if ϕ is a surjection of C−modules then ϕλ is also surjective. A similar result for injective maps is not true in general. However it is true in case ϕ is a monomorphism. We give an elementary proof for this case in the proposition 2. Let {e1, e2, . . . , em} is an order set of elements of E; T is a numbering of shape λ with elements in [m]. By replacing any i in a box of T by the element ei we have an element of E×λ. The image of this element in Eλ denoted by eT . By the functoriality of the construction of Eλ, any endomorphism of E determines an endomorphism of Eλ, this gives a left action of algebra EndC(E) on Eλ, vλ 7→ g  vλ such that the following diagram is commutative E×λ g×λ //  E×λ  Eλ g // Eλ. (∗∗) Proposition 3 give the exact formula for this action. In particular, the group GL(E) of automorphisms of E acts on the left on Eλ. Given a basis of E, one can identify E with Rm, then EndC(E) = MmC is the algebra of m × m matrices. Therefore MmC acts on Eλ, as does the subgroup GLm(C). So Schur module Eλ is a finite dimention polynomial representation of GLm(C). We refer the reader to [1] for the proof of irreducibility of Eλ. Acknowledgements. 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 Eλ if, in relation (3), one allows to inter- change between two adjacent columns only. Proof. Assume that for any v in E×λ, ϕ(v) = ∑ ϕ(w), where the sum is over all elements w in E×λ obtained from v 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, ϕ(v) = ∑ ϕ(w), 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 (l+ 2)nd column with lth column of v, (1 ≤ l ≤ λ1 − 2). Now let a1, a2, . . . , aµl be the entries of the l th column, respectively from the top to the bottom; y1, y2, . . . , yµl+1 be the entries of the (l+ 1) st column, respectively from the top to the bottom; and x1, x2, . . . , xµl+2 be the entries of the (l + 2) nd column, respectively from the top to the bottom, where k ≤ µl+2 ≤ µl+1 ≤ µl. This means that v = · · · a1 y1 x1 · · · a2 y2 x2 ... ... ... ... ... xµl+2 ... yµl+1 aµl . We have ϕ(v) = ∑ (j1,··· ,jk) ϕ ( · · · a1 ... yj1 · · · a2 x1 ... ... ... ylk ... ... xk+1 ... xk ... ... ... aµl ) = ∑ (j1,··· ,jk) ∑ (i1,··· ,ik) ϕ ( · · · ... ... yj1 · · · x1 ai1 ... ... ... yjk ... ... xk+1 xk aik ... ... ... ... ) = ∑ (i1,··· ,ik) ∑ (j1,··· ,jk) ϕ ( · · · ... ... yj1 · · · x1 ai1 ... ... ... yjk ... ... xk+1 xk aik ... ... ... ... ) = ∑ (i1,··· ,ik) ϕ ( · · · ... y1 ai1 · · · x1 y2 ... ... ... aik ... ... xk+1 xk ... ... ... yµl+1 ... ) , where (j1, . . . , jk) and (i1, . . . , ik) are k−tuples satisfied 1 6 j1 < j2 < · · · < jk 6 µl+1 and 1 6 i1 < i2 < · · · < ik 6 µl.  Proposition 2. Let ϕ : E → F be a homomorphism of finitely generated free C−modules. Then the following statements are equivalent (i) ϕ is a monomorphism; (ii) ϕλ : Eλ → F λ is a monomorphism for all λ; (iii) ϕλ is a monomorphism for some λ with at most m rows, m = rank(E). Proof. (i)⇒ (ii). Consider vλ ∈ kerϕλ, we have q(ϕ×λ(v)) = ϕλ(vλ) = 0 ∈ F λ ⇒ ϕ×λ(v) ∈ Qλ(F ). Since ϕ×λ is monomorphism then v ∈ Qλ(E), i.e., vλ is zero of Eλ. (ii)⇒ (iii). This is obvious. (iii)⇒ (i). Let {e1, e2, . . . , em} be a basis of E. We only need to show that ϕ(e1), ϕ(e2), . . . , ϕ(em) are linearly independent in F. Suppose that these elements are linearly dependent in F, i.e., we have a linear combination m∑ j=1 αjϕ(ej) = 0, where there exists at least a coefficient αj is not zero. Let j0 be the largest number such that αj0 6= 0. Let’s consider element v in E×λ defined by: +) If k ≥ j0 then v has entry at the first box of the jth0 row is m∑ j=1 αjej , and entries at the other boxes are ei if its box in the ith row. +) If k < j0 then v has entry at the first box of the bottom row is m∑ j=1 αjej , entries at different boxes of this row are ej0 , and entries at the other boxes are ei if its box in the ith row. For a such element v then q(ϕ×λ(v)) is zero in F×λ. In the other hand, it is clear that the statement vλ 6= 0 follows ϕλ(vλ) is not zero which contradicts the assumptions.  Proposition 3. Assume that g = (gij) is an element in MmC. If T has entries j1, j2, . . . , jn in its n boxes (ordered arbitrary), then g  eT = ∑ gi1j1 . . . ginjn .eT ′ , where the sum is taken over the mn fillings T ′ obtained from T by replacing the entries (j1, j2, . . . , jn) by (i1, i2, . . . , in), from [m]. Proof. Suppose that {e1, . . . , em} is a basis of E. We denote by e(T ), the element in E×λ which has image eT in Eλ. We have gej = m∑ i=1 gijei. Then g×λ(e(T )) = ∑ gi1j1 · · · ginjne(T ′)⇒ g  eT = ∑ gi1j1 · · · ginjneT ′ . By the diagram (**), the proof is complete.  References [1] W. Fulton, Young Tableaux - With Aplications to Representation Theory and Geometry, Cambridge University Press, 1997. [2] W. Fulton, J. Harris, Representation Theory - A first course, Graduate texts in mathematics, Springer- Verlag, New York, USA, 1991. [3] A. A. Kirillov, Elements of the Theory of Representations, Springer-Verlag, New York, USA, 1976.