Double Koszul Complex and Construction of Irreducible Representations of $\frak{gl}(3|1)$

The aim of this work is to give a combinatorial way to describe all irreducible representations in case the super-dimension of $V$ is $(3|1)$.


Preliminaries
This section presents some results on the general linear Lie super-algebras for the later use. We shall work with a field k of characteristic 0. A super vector space is a Z 2 -graded vector space V = V0 ⊕ V1. The spaces V0, V1 are called even and odd homogeneous components of V , their elements are called homogeneous. We denote the Z 2 -grade (or parity) of a homogeneous element a byâ. Assume dim V0 = m, dim V1 = n and fix a homogeneous basis of V : x 1 , . . . , x m ∈ V0, x m+1 , . . . , x m+n ∈ V1. For simplicity we denote the Z 2 -grade of x i byî. Thusî =0 if 1 ≤ i ≤ m andî =1 if m + 1 ≤ i ≤ m + n.
A Z 2 -graded algebra A is called a super algebra. Similarly we have the notion of super Lie algebra L, where the super anti-commutativity and the super Leibniz rule read: Here we use the convention that (−1)0 = 1 and (−1)1 = −1.
Given a super algebra A, the super-commutator on A, defined by C 0 define odd maps (i.e. maps that interchange the Z 2 -grading). An arbitrary map V → V is the sum of an even map with an odd map. This defines a Z 2 -grading on End(V ) and makes End(V ) a super algebra. The associated super Lie algebra End(V ) L is denoted by gl(V ).

2.2.
Representation of g = gl(V ). Let W be a super vector space. A super representation ρ of g in W is an even linear mapping ρ : g −→ gl(W ) which preserves the super commutator, that is a homomorphism of Lie super algebras. A super representation of g is also called a g-module. A super representation is said to be irreducible if it has no proper non-zero sub-representations. In oder to construct all irreducible representations of g we need the technique of induced representations, which we will now describe.

Induced representations.
A pair (U (g), i), where U (g) is an associative Z 2 -graded algebra and i : g −→ U (g) L is a homomorphism of Lie super algebra, is called a universal enveloping super algebra of g if for any other pair (U ′ , i ′ ), there is an unique homomorphism θ : U −→ U ′ such that i ′ = θ.i. Thus, the concepts of "super representation of g", "gmodule" and "left U (g)-module" are completely equivalent.
Let g be a super Lie algebra, U (g) be its universal enveloping super algebra, h be a super Lie sub-algebra of g, and V be an h-module. The Z 2 -graded space U (g) ⊗ U (h) V can be endowed with the structure of a g-module as follows: g(u ⊗ v) = gu ⊗ v for g ∈ g, u ∈ U (g), v ∈ V. The so constructed g-module is said to be induced from the h-module V and is denoted by Ind g h V .
2.2.2. Weights and Roots of g. The standard basis for g consists of matrices E ij : i, j = 1 . . . , m + n where E ij is the matrix with 1 in the place (i, j) and 0 elsewhere. Consider the sub-algebra h of g spanned by the elements h j := E jj : i, j = 1, . . . , m + n, h is a Cartan subalgebra of g. The space h * dual to h is is spanned by ǫ i : i = 1, . . . , m + n, where for Elements of h * are called the weights of g. Let λ ∈ h * , λ = m i+1 λ i ǫ i − m+n j=m+1 λ j ǫ j then we write λ = (λ 1 , λ 2 , . . . , λ m |λ m+1 , . . . , λ m+n ).
Then V (λ) is an irreducible representation with highest weight λ. The moduleV (λ) is called generalized Verma module or Kac module [8]. Kac showed that the V (λ)'s furnish all irreducible g-modules of finite dimension. If λ is typical weight then M λ = 0, thus V (λ) =V (λ), in this case V (λ) is called typical. On the other hand, if λ is atypical, an explicit construction of M (λ) is not known.

Characters of representations.
Let V be a finite-dimensional irreducible g-module.
For every element λ ∈ h * , we define The following formula for the character of typical irreducible modules is due to Kac [8]: with L 1 = α∈∆ + 1 (e α/2 + e −α/2 ), L 0 = β∈∆ + 0 (e β/2 − e −β/2 ). In [15], Su and Zhang gave an character formula for all finite dimension irreducible representations with any typical and atypical dominant integral weight λ. The formula is too complicated to recall here, but see below for a special case.

2.3.
Characters of irreducible representations of gl(3|1). In this section, we will recall formulas for the character of all typical and atypical finite-dimensional irreducible representations of gl(3|1). According to [15,Theorem 4.9].

Double Koszul complexes
3.1. The Koszul complex K. In [12] Manin suggested the following construction to define the super determinant of a super matrix. Let V * denote the vector space dual to V with the dual basic ξ 1 , ξ 2 , . . . , ξ d , ξ i (x j ) = δ i j . The complex K has its (k, l)-term given by K k,l := Λ k ⊗ S * l , where Λ k is the k-th homogeneous component of the exterior tensor algebra over V , S * l is the l-th homogeneous component of the symmetric tensor algebra over V * . The differential d k,l : K k,l −→ K k+1,l+1 is given by In fact, the construction above gives a series of complexes K a : here for k < 0 we define Λ k and S k to be 0. Thus each complex K a is bounded from below. It is easy to check that d k,l is gl(V )-equivariant; hence the homology groups of this complex are representations of gl(V ). On the other hand, one can show that the complex (K a , d) is exact everywhere if a = m − n, and the complex (K m−n , d) is exact everywhere except at the term Λ m ⊗S * n , where the homology group is one-dimensional. This homology group defines a one-dimensional representation of gl(V ). It turns out that elements of gl(V ) act on this representation by means of its super determinant.
Notice that there is another differential ∂ k,l : K k+1,l+1 −→ K k,l , which is defined as follows: One checks that on K k,l There is another Koszul complex associated to V . This complex was first defined by Priddy as a free resolution of k as a module over the symmetric tensor algebra of V (see [11]). As in the case of the complex K, the complex L with L p,r := S p ⊗ Λ r is defined as a series of complexes L a , with differential P p,r : L p,r −→ L p−1,r+1 given by The complexes (L • , P ) are exact, except for a = 0. We also have another differential Q p,r : L p−1,r+1 −→ L p,r , given by One checks that on L p,r r(p + 1)P Q + p(r + 1)QP = (p + r)id.
Consequently the complexes (L • , Q) are exact too.
3.3. The double Koszul complex. The main observation of this work is the fact that the two Koszul complexes mentioned in the previous section can be combined into a double complex which we call the double Koszul complex. In this section we describe this complex. An application to the construction of irreducible representations of the super Lie algebra gl(3|1) will be given in the next section.
For simplicity we shall use the dot "·" to denote the tensor product. Fix an integer a ≥ 1. We arrange the Koszul complexes K −a , K −a−1 , K −a−2 , . . . as in the diagram below.
To get the entries on a column into a complex we tensor each complex K −a−i with S i , i.e. the complex K −a−1 is tensored with S 1 , the complex K −a−2 is tensored with S 2 , etc. Then each column can be interpreted as the complexes L j tensored with S * a+j . Thus we have the following diagram where all rows are the Koszul complex K • tensored with S • and the columns are the Koszul complex L • tensored with S * • : A general square in diagram (8) has the form For convenience, we denote d := id ⊗ d, P := P ⊗ id. It is easy to show that P d = dP for all above squares.
We also have an exact double Koszul complex with d, P replaced by ∂, Q.
The commutativity of this diagram is easy to check.
3.4. Some remarks on the structure of the double complex. In this subsection we study some maps obtained from the differentials of the double Koszul complex. From now, we only consider the case (m|n) = (3|1).
We put the two diagrams (8) and (10) into one: Proof. According to formulas (6) and (7) and the commutativity between d, P and ∂, Q, we have Qd∂P.
We will use induction on i to prove that ∂P Qd : S i · S * a+i −→ S i · S * a+i is diagonalizable with the set of eigenvalues A i := (a + i + 3 − j)j (i + 1)(a + i + 1) , j = 1, 2, . . . , i + 1 .
For i = 0 the claim follows from the equation above. Assume that the proposition is true for i − 1.
By assumption ∂P Qd : Consider the diagram in (8) as an exact sequence of horizontal complexes (except for the first column) and split it into short exact sequences.
We have P d∂Q = dP Q∂ and this operator can be restricted to KerP 1,k+1 · S * a+k+2 . We compute the eigenvalue of this operator. First we have Notice that dQP ∂ is an endomorphism of Imd ⊂ S i · Λ k+1 · S * a+i+k+1 , on this space d∂ operates by multiplication with a+3 (k+1)(a+k+2) . On the other hand from above we know the eigenvalues of P ∂dQ form the set A 0 . Thus dQP ∂ is diagonalizable with eigenvalues A 0 ∪ {0}. Consequently dP Q∂ is diagonalizable with eigenvalues a + 3 2k(a + k + 2) , a + 2 2(k + 1)(a + k + 2) , 0 .
On the other hand, the restriction of P Q to KerP 1,k+1 · S * a+k+2 is the multiplication with k+2 2(k+1) . Therefore, the eigenvalues of P ∂dQ are , .
In general, we consider the composed map dQP ∂.

Construction of irreducible representations of gl(V ).
Let V be a super vector space with super-dimension (3|1). In this section, using the double Koszul complex, we will construct all irreducible representations of this super algebra. To show the representations obtained are in fact irreducible we compute their characters.

4.1.
Combinatorial construction of irreducible representations of gl(V ). In this section, we will compute the character of the duals of irreducible direct summand of the power of the fundamental representation V . By the combinatorial method, we have where I λ are simple, and Γ 3,1 is the set of partitions with λ 4 ≤ 1. Since the character of V is x 1 + x 2 + x 3 − y, using the determinant formula (3.5) of [10], we can compute the character of I λ for all λ ∈ Γ 3,1 .

4.2.
Construct representations by using Koszul complex K. Consider complexes K a , with a := k − l = 2.
..., By using the exactness property of the Koszul complex K, we will construct a class of irreducible representations of gl(3|1). According to (6) we have Consequently, we have ( [3]).