Preliminaries

We begin with the definition of the \(n\)-th tensor product.

\(n\)-th tensor product

Definition. For a finite dimensional complex vector space \(V\), and an natural number \(n\), we define the complex \(n\)-th tensor product as follows \[ V^{\otimes n}:=F\left(V^{n}\right) / \sim \] where \(F\left(V^{n}\right)\) is the free abelian group on \(V^{n}\), considered as a set, and \(\sim\) is the equivalence relation defined for \(v_{1}, \ldots, v_{n}, \hat{v}_{1}, \ldots, \hat{v}_{n}, w_{1}, \ldots, w_{n} \in V\), as \[ \left(v_{1}, \ldots, v_{n}\right) \sim\left(w_{1}, \ldots, w_{n}\right) \] whenever there exists a \(k \in \mathbb{C}\) such that \(k\left(v_{1}, . ., v_{n}\right)=\left(w_{1}, \ldots, w_{n}\right)\), and \[ \left(v_{1}, . ., v_{i}, \ldots v_{n}\right)+\left(v_{1}, \ldots, \hat{v}_{i}, \ldots, v_{n}\right) \sim\left(v_{1}, . ., v_{i}+\hat{v}_{i}, . ., v_{n}\right) \]

Remark. One can easily verify that this is indeed an equivalence relation.

Next we will use the following definitions of some interesting subvector spaces of the \(n\)-th tensor product, which will be of the most interest in this paper.

\(n\)-th symmetric and alternating spaces (powers)

Definition. For a complex finite dimensional vector space \(V\), and a natural number \(n\), we define the complex \(n\)-th symmetric space denoted \(\operatorname{Sym}^{n}(V)\), as the subspace of \(V^{\otimes n}\), spanned by the following set, \[ \left\{\sum_{\sigma \in \mathfrak{S}_{n}} v_{\sigma(1)} \otimes \ldots \otimes v_{\sigma(n)} \mid v_{i} \in V\right\} \] and denote a generator \(\sum\limits_{\sigma \in \mathfrak{S}_{n}} v_{\sigma(1)} \otimes \ldots \otimes v_{\sigma(n)}=v_{1} \cdots v_{n}\).

Definition. For a finite dimensional complex vector space \(V\), and an natural number \(n\), we define the complex \(n\)-th alternating space denoted \(\bigwedge^{n} V\), as the subspace of \(V^{\otimes n}\), spanned by the following set \[ \left\{\sum_{\sigma \in \mathfrak{S}_{n}} \operatorname{sgn}(\sigma) v_{\sigma(1)} \otimes \ldots \otimes v_{\sigma(n)} \mid v_{i} \in V\right\} \] and denote a generator \(\sum_{\sigma \in \mathfrak{S}_{n}} \operatorname{sgn}(\sigma) v_{\sigma(1)} \otimes \ldots \otimes v_{\sigma(n)}=v_{1} \wedge \cdots \wedge v_{n}\).

Remark. The experienced and curious reader may be asking themselves why we chose these definitions, rather than more accepted ones, while these definitions, over \(\mathbb{C}\), are equivalent it takes some work to show this non-trivial fact, so to avoid the discussion we chose these definitions.

Complex Group Algebra

Definition. For a group \(G\), we define the complex group algebra of \(G\) denoted as \(\mathbb{C} G\) as, the \(|G|\)-dimensional complex vector space with the canonical basis indexed by elements of \(G\), that is for \(g \in G\) we have a basis element \(e_{g} \in \mathbb{C} G\), where we define the multiplication in \(\mathbb{C} G\) on basis elements, for \(g_{1}, g_{2} \in G\) as \[ e_{g_{1}} \cdot e_{g_{2}}=e_{g_{1} \cdot g_{2}} \] and expand the definition linearly for the remaining elements.

Remark. We can see that if \(G\) is non-abelian then so is \(\mathbb{C} G\) and if \(|G|=\infty\) then \(\mathbb{C} G\) has infinite dimension.

Combinatorics

Let \( A = (A_0, A_1) \) be a \(\mathbb{Z}_2\)-graded set, i.e. the pair of sets indexed by \(\{0, 1\}\). Assume that the set \( A \) is ordered by a total order \( \triangleleft \). A tableau of shape \(\lambda / \mu\) with values in \( A \) is a function \( T : D(\lambda / \mu) \rightarrow A \).

Definition (Row Standard). A tableau \( T \) of shape \(\lambda / \mu\) with values in \( A \) is row standard if for each \((u, v)\) we have \( T(u, v) \triangleleft T(u, v + 1) \) with equality possible if \( T(u, v) \in A_1 \).

Definition (Column Standard). We say that a tableau \( T \) of shape \(\lambda / \mu\) with values in \( A \) is column standard if \( T(u, v) \triangleleft T(u + 1, v) \) with equality possible when \( T(u, v) \in A_0 \).

Definition (Standard). A tableau \( T \) of shape \(\lambda / \mu\) with values in \( A \) is standard if it is both column standard and row standard.

Notation.

Reference. "Cohomology of Vector Bundles and Syzygies". Page 10.

Let \(\lambda / \mu\) be a skew partition, and let \( A = (A_0, A_1) \) be a \(\mathbb{Z}_2\)-graded set ordered by the total order \( \triangleleft \). We define the orders \( \preceq \) (relative to \( \triangleleft \)) on the sets of row standard (column standard, standard) tableaux as follows.

\( \preceq \) on \( \text{RST}(\lambda / \mu, A) \)

Given two tableaux \( T, U \in \text{RST}(\lambda / \mu, A)\), we have \( T \preceq U \) if \( T = U \). Assume that \( T \neq U \). Let us write them as \( T = (T_1, \ldots, T_s) \), \( U = (U_1, \ldots, U_s) \) with \( T_i \) being the part of \( T \) from the \( i \)-th row of \(\lambda / \mu\), and \( U_i \) being the part of \( U \) from the \( i \)-th row of \(\lambda / \mu\). Let \( j \) be the minimal \( i \) for which \( T_i \neq U_i \). We have \( T_j = (T(j, 1), \ldots, T(j, \lambda_j - \mu_j)) \), \( U_j = (U(j, 1), \ldots, U(j, \lambda_j - \mu_j)) \). Now let \( k \) be the smallest index for which \( T(j, k) \neq U(j, k) \) (such a \( k \) exists by the choice of \( j \)). We say that \( T \preceq U \) if and only if \( T(j, k) \triangleleft U(j, k) \).

\( \preceq \) on \( \text{ST}(\lambda / \mu, A) \)

The order \( \preceq \) on \( \text{ST}(\lambda / \mu, A) \) is defined to be the restriction of \( \preceq \) from \( \text{RST}(\lambda / \mu, A) \).

\( \preceq \) on \( \text{CST}(\lambda / \mu, A) \)

Finally we define the order \( \preceq \) on \( \text{CST}(\lambda / \mu, A) \). Given two tableaux \( T, U \) from \( \text{CST}(\lambda / \mu, A) \), then \( T = U \) implies \( T \preceq U \). Assume \( T \neq U \). We write \( T = (T^1, \ldots, T^s) \), \( U = (U^1, \ldots, U^s) \) with \( T^i \) being the part of \( T \) from the \( i \)-th column of \(\lambda / \mu\), and \( U^i \) being the part of \( U \) from the \( i \)-th column of \(\lambda / \mu\). Let \( j \) be the minimal \( i \) for which \( T^i \neq U^i \). We have \( T^j = (T(1, j), \ldots, T(\lambda'_j - \mu'_j, j)) \), \( U^j = (U(1, j), \ldots, U(\lambda'_j - \mu'_j, j)) \). Now let \( k \) be the smallest index for which \( T(k, j) \neq U(k, j) \) (such \( k \) exists by the choice of \( j \)). We say that \( T \preceq U \) if and only if \( T(k, j) \triangleleft U(k, j) \).

Reference. "Cohomology of Vector Bundles and Syzygies". Page 11.

Schur Module

In this part we describe some uses of tableaux in studying representations of the symmetric group \(S_n\) and the general linear group \(G L_m(\mathbb{C})\). We will see that to each partition \(\lambda\) of \(n\) one can construct an irreducible representation \(S^\lambda\) of the symmetric group \(S_n\) (called a Specht module) and an irreducible representation \(E^\lambda\) of \(G L(E)\) for \(E\) a finite dimensional complex vector space (called a Schur or Weyl module). The space \(S^\lambda\) will have a basis with one element \(v_T\) for each standard tableau \(T\) of shape \(\lambda\). If \(e_1, \ldots, e_m\) is a basis for \(E\), then \(E^\lambda\) will have a basis with one element \(e_T\) for each (semistandard) tableau \(T\) on \(\lambda\) with entries from \([ m ]\). These basis vectors \(e_T\) will be eigenvectors for the diagonal matrix with entries \(x_1, \ldots, x_m\), with eigenvalue \(x^T\); the character of the representation will be the Schur polynomial \(s_\lambda\left(x_1, \ldots, x_m\right)\).

Notation \(E^{\times n}\)

Remark. The notation \(E^{\times n}\) denotes the \(n\)-fold Cartesian product of the \(R\)-module \(E\) with itself. This means that \(E^{\times n}\) is the set of all ordered \(n\)-tuples \((e_1, e_2, \ldots, e_n)\), where each \(e_i\) is an element of \(E\). In other words, \[ E^{\times n} = E \times E \times \cdots \times E \quad (\text{\(n\) times}). \]

Notation \(E^{\times \lambda}\)

Remark. We will write $E^{\times \lambda}$ for the cartesian product of $n=|\lambda|$ copies of $E$, but labelled by the $n$ boxes of (the Young diagram of) $\lambda$. So an element $\mathbf{v}$ of $E^{\times \lambda}$ is given by specifying an element of $E$ for each box in $\lambda$.

Example. Let \(E = \mathbb{C}^2\) and \(\lambda = (2)\). We can know the young diagram of \(\lambda\) is \[ \lambda = (2) \;=\; \begin{array}{|l|l|l|} \hline (1, 1) & (1, 2) \\ \hline \end{array} \] Here, we can know that \(E^{\times \lambda}=\{v_{1, 1}, v_{1, 2}\}\), where \(v_{i, j}\) is indexed with \((i, j)\) coordinate of the young diagram and \(v_{i, j}\) is an element of \(E\).

Exchange

In this part, we need the notion of an exchange. This depends on a choice of two columns of a Young diagram \(\lambda\), and a choice of a set of the same number of boxes in each column. For any filling \(T\) of $\lambda$ (with entries in any set), the corresponding exchange is the filling \(S\) obtained from \(T\) by interchanging the entries in the two chosen sets of boxes, maintaining the vertical order in each; the entries outside these chosen boxes are unchanged. For example, if \(\lambda=(4,3,3,2)\), and the chosen boxes are the top two in the third column, and the second and fourth box in the second column, the exchange takes

Need an exchange example here.

Reference. "Young Tableaux : With Applications to Representation Theory and Geometry". Page 81.

Map \(\varphi\)

\(R\)-Multilinearity Map

Definition. A multilinear map is a function \[ f: V_1 \times \cdots \times V_n \rightarrow W, \] where \(V_1, \ldots, V_n\left(n \in \mathbb{Z}_{\geq 0}\right)\) and \(W\) are vector spaces (or modules over a commutative ring), with the following property: for each \(i\), if all of the variables but \(v_i\) are held constant, then \(f\left(v_1, \ldots, v_i, \ldots, v_n\right)\) is a linear function of \(v_i\).

Example. A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer \(k\), a multilinear map of \(k\) variables is called a \(k\)-linear map.

Return to our context, imagine \( E^{\times \lambda} \) as a Cartesian product of \( n \) copies of \( E \), where \( n = |\lambda| \) is the total number of boxes in the Young diagram of \( \lambda \). An element \( \mathbf{v} \in E^{\times \lambda} \) is a tuple \( (v_1, v_2, \dots, v_n) \), and \( \varphi(\mathbf{v}) \) depends on all these entries. Multilinearity means:

For further information, please check wikipedia, Multilinear Map.

We define the maps \( \varphi: E^{\times \lambda} \rightarrow F \) from \( E^{\times \lambda} \) to an \( R \)-module \( F \), satisfying the following three properties:

1. \( \varphi \) is \( R \)-multilinear.

This means that if all the entries but one are fixed, then \( \varphi \) is \( R \)-linear in that entry.

Example. See \(R\)-Multilinearity Map.

2. \( \varphi \) is alternating in the entries of any column of \( \lambda \).

That is, \( \varphi \) vanishes whenever two entries in the same column are equal.

Example: Let \(\lambda = (2, 1)\), with Young diagram. Assign \(\mathbf{v} = \{v_{1,1}, v_{1,2}, v_{2,1}\}\), where \(v_{1,1}, v_{1,2}, v_{2,1} \in E\). Suppose that \(v_{1,1} = v_{2,1}\) (be aware that those two entries are on the same column), then we get \(\varphi(\mathbf{v}) = 0\).

This property introduces an “antisymmetric” behavior within each column, which is critical for constructing irreducible representations.

3. For any \( \mathbf{v} \) in \( E^{\times \lambda} \), \(\varphi(\mathbf{v}) = \sum \varphi(\mathbf{w})\), where the sum is over all \( \mathbf{w} \) obtained from \( \mathbf{v} \) by an exchange between two given columns, with a given subset of boxes in the right chosen column.

Example: Let \(\lambda = (3, 2)\), with Young diagram: Suppose we choose the first and second columns and focus on the two boxes in each column. Let \(\mathbf{v} = \{v_{1,1}, v_{1,2}, v_{1,3}, v_{2,1}, v_{2,2}\}\), where each \(v_{i,j} \in E\).

• Perform exchanges: 1. Swap \(v_{1,1} \leftrightarrow v_{2,1}\), keeping other entries fixed. 2. Swap \(v_{1,2} \leftrightarrow v_{2,2}\). The map \(\varphi(\mathbf{v})\) sums over all such exchanges, with appropriate sign changes (based on permutation parity).

Schur Module

Definition. We define the Schur module \(E^\lambda\) to be the universal target module for such maps \(\varphi\).

Remark. This means that \(E^\lambda\) is an \(R\)-module, and we have a map \(E^{\times \lambda} \rightarrow E^\lambda\), that we denote \(\mathbf{v} \mapsto \mathbf{v}^\lambda\), satisfying (1)-(3), and such that for any \(\varphi: E^{\times \lambda} \rightarrow F\) satisfying (1)-(3), there is a unique homomorphism \(\widetilde{\varphi}: E^\lambda \rightarrow F\) of \(R\)-modules such that \(\varphi(\mathbf{v})=\widetilde{\varphi}\left(\mathbf{v}^\lambda\right)\) for all \(\mathbf{v}\) in \(E^{\times \lambda}\).

Example: Schur Module for \(\lambda = (2)\) : Let \(R = \mathbb{R}\) and \(E = \mathbb{R}^2\). The Young diagram for \(\lambda = (2)\) has two boxes in one row: \[ \begin{array}{|l|l|} \hline (1, 1) & (1, 2) \\ \hline \end{array}. \] • \(E^{\times \lambda} = E \times E = \mathbb{R}^2 \times \mathbb{R}^2\) consists of all pairs \((v_1, v_2)\) of vectors in \(E\). • \(E^\lambda\) imposes antisymmetry on columns (though trivial here since \(\lambda = (2)\) has no columns) and handles exchange invariance. • If \(\varphi(v_1, v_2) = v_1 - v_2\), then \(E^\lambda\) is the submodule encoding this alternating behavior.

Reference. "Young Tableaux : With Applications to Representation Theory and Geometry". Page 105.

Example. For one extreme, when \(\lambda = (n)\), the Schur module \(E^{(n)}\) is the \(n^{\text{th}}\) symmetric power \(\operatorname{Sym}^nE\).

Reference. "Young Tableaux : With Applications to Representation Theory and Geometry". Page 79.

Example. For the other extreme, when \(\lambda = (1^n)\), the Schur module \(E^\lambda\) is the \(n^{\text{th}}\) exterior power \(\bigwedge^n(E)\).

Reference. "Young Tableaux : With Applications to Representation Theory and Geometry". Page 80.

Weyman's Construction

Now, we explore Weyman's way to construct Schur Module.

Few things to keep in mind:

Weyman called "young's diagram" as "young frame", and he firstly introduced it in his book "Cohomology of Vector Bundles and Syzygies" on page 8.

Another uncommon thing is Weyman using \(\Sigma_r\) to denote the symmetric group of degree \(r\), which is usually denoted by \(S_r\).

Definition. Let \(e_1, \ldots, e_n\) will be a fixed ordered basis of \(E\). We introduce the ordered set \([1, n]=\{1,2, \ldots, n\}\), which is the set indexing our basis. We refer to section 1.1.2. for notions related to tableaux used in this section. Let \(T\) be a tableau of shape \(\lambda\) with the entries in \([1, n]\). We associate to \(T\) the element in \(L_\lambda E\) which is a coset of the tensor \[ \begin{aligned} & e_{T(1,1)} \wedge \ldots \wedge e_{T\left(1, \lambda_1\right)} \otimes e_{T(2,1)} \wedge \ldots \wedge e_{T\left(2, \lambda_2\right)} \\ & \quad \otimes \ldots \otimes e_{T(s, 1)} \wedge \ldots \wedge e_{T\left(s, \lambda_s\right)} \end{aligned} \] in \(L_\lambda E\). In the sequel we will identify these two objects and we will call both of them the tableaux of shape \(\lambda\) corresponding to the basis \(\left\{e_1, e_2, \ldots, e_n\right\}\).

1. Basic Setup

1.1. Free Module \(E\)

Definition: Let \(E\) be a free module of dimension \(n\) over a commutative ring \(\mathbf{K}\).

1.2. Partition \(\lambda\)

Definition: Let \(\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_s)\) be a partition of a positive integer \(m\).

1.3. Young Scheme of Partition \(\lambda\)

Definition: In general, the Young scheme of a partition \(\lambda\) is a young diagram of shape \(\lambda\) with some boxes empty and some filled.

Reference."Cohomology of Vector Bundles and Syzygies", Page 33.

1.4. Symmetric Powers

Definition. The \( r \)-th symmetric power \( S_rE \) of \( E \) to be the \( r \)-th tensor power \( E^{\otimes r} \) of \( E \) divided by the submodule generated by the elements \[ u_1 \otimes \cdots \otimes u_r - u_{\sigma(1)} \otimes \cdots \otimes u_{\sigma(r)} \] for all \( \sigma \in \Sigma_r, \, u_1, \ldots, u_r \in E \). We denote the coset of \( u_1 \otimes \cdots \otimes u_r \) by \( u_1 \cdots u_r \).

Reference."Cohomology of Vector Bundles and Syzygies", Page 3.

1.5. Exterior Powers

We define the \( r \)-th exterior power \( \bigwedge^r E \) of \( E \) to be the \( r \)-th tensor power \( E^{\otimes r} \) of \( E \) divided by the submodule generated by the elements: \[ u_1 \otimes \cdots \otimes u_r - (-1)^{\operatorname{sgn} \sigma} u_{\sigma(1)} \otimes \cdots \otimes u_{\sigma(r)} \] for all \( \sigma \in \Sigma_r, \, u_1, \ldots, u_r \in E \). We denote the coset of \( u_1 \otimes \cdots \otimes u_r \) by \( u_1 \wedge \cdots \wedge u_r \).

Reference."Cohomology of Vector Bundles and Syzygies", Page 1.

1.6. Diagonal Map \(\Delta\)

The components of \( \Delta \) will be denoted by \( \Delta : \bigwedge^{r+s} E \to \bigwedge^r E \otimes \bigwedge^s E \). In terms of elements we have

Reference."Cohomology of Vector Bundles and Syzygies", Page 3.

1.7 Multiplication Map \(m\)

Definition. For exterior algebras, the multiplication map \( m \) is defined as \(m: \bigwedge^k E \otimes \bigwedge^l E \rightarrow \bigwedge^{k+l} E \) by

2. Constructing the Module \(L_\lambda E\)

2.1. Tensor Product of Exterior Powers

Construction: Form the tensor product of the exterior powers based on the partition \(\lambda\):

Purpose: This tensor product serves as the foundational structure before imposing relations.

2.2. Defining the Maps \(\theta(\lambda, a, u, v; E)\)

Purpose: These maps impose relations that ensure the resulting module \(L_\lambda E\) captures the desired representation-theoretic properties.

Definition of \(\theta(\lambda, a, u, v; E)\):

Geometric Interpretation:

2.3. Submodule \(R_{a, a+1}(E)\):

Definition: For each adjacent pair of exterior powers indexed by \(a\) and \(a+1\), \(R_{a, a+1}(E)\) is a submodule spanned by the images of specific maps \(\theta(\lambda, a, u, v; E)\).
Condition: These maps are considered only when \(u + v \lt \lambda_{a+1}\).

In other words,

2.4. Introducing Relations via Submodule \(R(\lambda, E)\)

Definition of \(R(\lambda, E)\): Suppose that the length of a partition \(\lambda\) is \(s\). Then, the submodule \(R(\lambda, E)\) is defined as

2.5. Construction of \(L_\lambda E\)

Definition: \[ L_\lambda E = \frac{\bigwedge^{\lambda_1} E \otimes \bigwedge^{\lambda_2} E \otimes \ldots \otimes \bigwedge^{\lambda_s} E}{R(\lambda, E)} \]

Explanation: By quotienting out the submodule \(R(\lambda, E)\), we impose the necessary relations to form the Schur module \(L_\lambda E\).

2.6.1. Setting a Partition and Parameters

Let's take a simple partition \(\lambda = (2, 1)\). Here, \(\lambda_1 = 2\), \(\lambda_2 = 1\) and the length \(s = 2\).

Suppose \(E\) is a free module over a commutative ring \(R\), say \(E = R^3\) for simplicity.

3.2. Identifying Adjacent Exterior Powers

Given \(\lambda = (2, 1)\), there is only one adjacent pair (\(a = 1\)):

• \(\bigwedge^{\lambda_1} E = \bigwedge^2 E\)
• \(\bigwedge^{\lambda_2} E = \bigwedge^1 E = E\)

3.3. Defining the \(\theta\) Maps

For \(a = 1\), we define \(\theta(\lambda, 1, u, v; E)\) maps where \(u + v \lt \lambda_{a+1} = \lambda_2 = 1\).

Given that \(u + v \lt 1\), the only possible values are:

• \(u = 0, v = 0\) (since \(u, v \geq 0\) as tensor degrees are non-negative integers)

Thus, the map simplifies to:

\(\theta(\lambda, 1, 0, 0; E): \bigwedge^0 E \otimes \bigwedge^{2 - 0 + 1 - 0} E \otimes \bigwedge^0 E \rightarrow \bigwedge^2 E \otimes \bigwedge^1 E\)

Simplifying further:

\(\theta(\lambda, 1, 0, 0; E): R \otimes \bigwedge^3 E \otimes R \rightarrow \bigwedge^2 E \otimes E\)

Here, \(\bigwedge^0 E = R\) (since \(\bigwedge^0 E\) is the ground ring itself).

The map can be interpreted as:

\(\theta: R \otimes \bigwedge^3 E \otimes R \rightarrow \bigwedge^2 E \otimes E\)

Given the tensor product with \(R\) (scalars), the map essentially collapses the tensor product to:

\(\theta(r_1, \omega, r_2) = r_1 r_2 \cdot \omega\)

However, in this specific case, since \(\bigwedge^3 E\) maps to \(\bigwedge^2 E \otimes E\), the map likely involves a contraction or splitting of the 3-form into a 2-form and a 1-form, respecting the module structure.

3.4. Constructing \(R_{1, 2}(E)\)

The submodule \(R_{1, 2}(E)\) is spanned by the image of \(\theta(\lambda, 1, 0, 0; E)\):

\(R_{1, 2}(E) = \text{Span}\left\{ \theta(\lambda, 1, 0, 0; E)(x) \mid x \in \bigwedge^0 E \otimes \bigwedge^3 E \otimes \bigwedge^0 E \right\}\)

Given our simplified \(\theta\) map, \(R_{1, 2}(E)\) consists of all elements of the form \(r \cdot \omega\), where \(r \in R\) and \(\omega \in \bigwedge^3 E\), mapped into \(\bigwedge^2 E \otimes E\).

3.5. Final Construction of \(L_\lambda E\)

Now, \(L_\lambda E\) is:

\(L_\lambda E = \frac{\bigwedge^2 E \otimes E}{R(\lambda, E)} = \frac{\bigwedge^2 E \otimes E}{R_{1, 2}(E)}\)

3. Basis and Indexing

3.1. Fixed Ordered Basis of \(E\)

Notation: Let \(e_1, e_2, \ldots, e_n\) be a fixed ordered basis of \(E\).

Purpose: This basis allows us to concretely represent elements in \(L_\lambda E\).

3.2. Index Set \([1, n]\)

Definition: Define the ordered set \([1, n] = \{1, 2, \ldots, n\}\).

Usage: This set indexes the basis elements of \(E\), facilitating the association with tableaux.

Remark. The notation of \([1, n]\) is the same of the notation of \([n]\) in combinatorics.

4. Associating Tableaux to Elements in \(L_\lambda E\)

4.1. Tableau \(T\) of Shape \(\lambda\)

Definition: Let \(T\) be a tableau of shape \(\lambda\) with entries from the set \([1, n]\).

4.2. Mapping Tableau to \(L_\lambda E\)

Association:

To each tableau \(T\), associate the element in \(L_\lambda E\) represented by the coset of the following tensor product:

Explanation: Each row of the tableau corresponds to an exterior power, and the entries determine which basis elements are wedged together.

4.3. Identification of Objects

Statement: In \(L_\lambda E\), we identify the tableau \(T\) with its corresponding tensor product element.

Implication: This identification allows us to treat tableaux as concrete elements within the module \(L_\lambda E\), facilitating computations and further theoretical developments.

Reference. "Cohomology of Vector Bundles and Syzygies". Page 32.

5. Combinatorial Interpretation

(2.1.1) Remark. It is convenient to think about the relations \( R(\lambda, E) \) in graphical terms using the Young frames.

Young Scheme

Definition. The Young scheme of shape \( \lambda \) to be the Young frame of shape \( \lambda \) with some boxes empty and some filled. We associate to the map \( \theta(\lambda, a, u, v; E) \) its Young scheme which is empty in all rows except the \( a \)-th and \( (a + 1) \)-st, and its restriction to these rows is with \( u \) empty boxes, followed by \( \lambda_a - u \) filled boxes in the \( a \)-th row, and \( \lambda_{a+1} - v \) filled boxes, followed by \( v \) empty boxes in the \( (a + 1) \)-st row.

Notice that the condition \( u + v \lt \lambda_{a+1} \) assures that there will be at least one column in this frame with two boxes filled.

Example. Take \( \lambda = (3, 3), u = v = 1 \). We firstly have the tensor product of the following:

Definition. The image of typical element:

Reference. "Cohomology of Vector Bundles and Syzygies". Page 32.

Example. Follow the previous example, we have \( \lambda = (3, 3), u = v = 1 \). We firstly have Young scheme as

Remark. The term "exterior diagonal" refers to the process of keeping track of the sign changes that occur due to the antisymmetry of the wedge product as we perform the shuffling (permutations).

Wedge Product and Antisymmetry: The wedge product (\(\bigwedge\)) is a bilinear, alternating product. The key property for our discussion is antisymmetry:

Tableaux and Wedge Products: Imagine each row of a tableau as representing a wedge product of basis vectors. For example, if a row has elements a, b, and c, it corresponds to eₐ ∧ eь ∧ e𝒸.

Shuffling and Permutations: When we "shuffle" the \(y_i\) elements into the filled boxes, we are effectively permuting the elements within these wedge products. Each swap of two elements corresponds to a multiplication by \(-1\) due to the antisymmetry of the wedge product.

Remark. Start with a reference tableau (where the \(y_i\) are in some standard order, like increasing order from left to right and top to bottom). This tableau has a positive sign by convention.

Example. If \(\lambda = (t)\) then \(L_\lambda E = \bigwedge^t E\). Indeed, by definition \(L_\lambda E = \bigwedge^t E / R((t), E)\), but \(R((t), E) = 0\), since the partition \((t)\) has only one part.

Example. If \(\lambda = (1^t)\) then \(L_\lambda E = S_t E\). Indeed, the relations \(R_{a,a+1}(E)\) express the symmetry between the \(a\)-th and \((a + 1)\)st row.

We want to go through the above example in detail. Firstly, we have

Example. If \(\lambda = (2, 1)\) then \(\lambda_1 = 2, \lambda_2 = 1\). Suppose that \(\lambda_a = 2, \lambda_{a+1} = 1\). Based on the condition that \(u + v \lt \lambda_{a+1}\), we have \(u = v = 0\).

Definition. It follows that the Schur functor \( L_{(p, 1^{q-1})} E \) is the cokernel of the composition map

Reference. "Cohomology of Vector Bundles and Syzygies". Page 36.

Schur module corresponding to the partition \(\lambda\).

Reference. "Cohomology of Vector Bundles and Syzygies". Page 39.

6. Main Theorems

(6.1.) Theorem. Let \( E \) be a free \( \mathbb{K} \)-module of dimension \( n \). Let \( e_1, \ldots, e_n \) be a basis of \( E \). The set \( \text{ST}(\lambda, [1, n]) \) of standard tableaux of shape \( \lambda \) with entries from \( [1, n] \) form a basis of \( L_\lambda E \). In particular, \( L_\lambda E \) is also a free module.

Proof. First we prove that the standard tableaux generate \( L_\lambda E \). It is clear that the set \( \text{RST}(\lambda, [1, n]) \) of row standard tableaux with entries from \([1, n]\) generates \( L_\lambda E \). Let us order the set of such tableaux by the order \( \preceq \) defined in section 1.1.2. We will prove that if the tableau \( T \) is not standard then we can express it modulo \( R(\lambda, E) \) as a combination of earlier tableaux. Let us assume first that \( T \) has two rows. Since \( T \) is not standard, we can find \( w \) for which \( T(1, w) > T(2, w) \). We consider the map \(\theta(\lambda, 1, u, v; E)\) for \( u = w - 1 \) and \( v = \lambda_2 - w \). The key observation is that image of the tensor \( V_1 \otimes V_2 \otimes V_3 \), where \[ \begin{aligned} V_1 &= e_{T(1,1)} \wedge e_{T(1,2)} \wedge \ldots \wedge e_{T(1,w-1)}, \\ V_2 &= e_{T(1,w)} \wedge \ldots \wedge e_{T(1,\lambda_1)} \wedge e_{T(2,1)} \wedge \ldots \wedge e_{T(2,w)}, \\ V_3 &= e_{T(2,w+1)} \wedge \ldots \wedge e_{T(2,\lambda_2)}, \end{aligned} \] contains the tableau \( T \) with the coefficient 1, and all the other tableaux occurring in this image are earlier than \( T \) in the order \( \leq \). Indeed, in all summands other than \( T \) we shuffle the smaller numbers from the second row to the first one, replacing bigger numbers. Therefore \( T \) can be expressed modulo \( R(\lambda, E) \) as a combination of earlier tableaux. Let us consider the general case. If \( T \) is not standard, then we can find such \( a \) and \( w \) that \( T(a, w) > T(a + 1, w) \). Now we apply the previous argument to the tableau \( S \) which consists of the \( a \)-th and \((a+1)\)st rows of \( T \). Notice that the relations \( R(\lambda, E) \) we are using do not do anything to the other rows of \( T \), so we can express \( T \) as a sum of earlier tableaux in the order \( \preceq \). It remains to prove that the standard tableaux are linearly independent in \( L_\lambda E \). Consider a map \[ \phi_\lambda : \bigwedge^{\lambda_1} E \otimes \bigwedge^{\lambda_2} E \otimes \ldots \otimes \bigwedge^{\lambda_s} E \xrightarrow{\alpha} \bigotimes_{(i,j) \in \lambda} E(i,j) \] \[ \xrightarrow{\beta} S_{\lambda'_1} E \otimes S_{\lambda'_2} E \otimes \ldots \otimes S_{\lambda'_t} E, \] where \( \alpha \) is the tensor product of exterior diagonals \[ \Delta : \bigwedge^{\lambda_j} E \rightarrow E(j,1) \otimes E(j,2) \otimes \ldots \otimes E(j,\lambda'_j) \] and \( \beta \) is the tensor product of multiplications \[ m : E(1,i) \otimes E(2,i) \otimes \ldots \otimes E(\lambda'_i,i) \rightarrow S_{\lambda'_i} E. \] If we imagine the copies of \( E \) correspond to the boxes of \( \lambda \) with \( \bigwedge^{\lambda_j} E \) corresponding to the boxes in the first row and \( S_{\lambda'_i} E \) corresponding to boxes in the \( i \)-th column of \( \lambda \), we can think of the image \( \phi_\lambda(T) \) of the tableau \( T \) as first shuffling (with signs) the terms of \( T \) in each row, and then multiplying the terms in each column of the tableau we obtain. The map \( \phi_\lambda \) is called the Schur map associated to the partition \( \lambda \).

Why is it "clear" that the row-standard tableaux span \(L_\lambda E\)? Essentially, each row of a tableau in \(L_\lambda E\) corresponds to an element \[ e_{i_1} \wedge e_{i_2} \wedge \cdots \wedge e_{i_{\lambda_a}} \] in an exterior power \(\bigwedge^{\lambda_a} E\). But a well-known fact is that \(\bigwedge^r E\) is spanned by wedges of the form \[ e_{j_1} \wedge e_{j_2} \wedge \cdots \wedge e_{j_r} \] with \(j_1 \lt j_2 \lt \cdots \lt j_r\), i.e. the indices are strictly increasing with respect to the total order on \(\{1,2,\dots,n\}\). This exactly says that, up to a possible sign, one may reorder any wedge \(\,e_{i_1}\wedge e_{i_2}\wedge \cdots \wedge e_{i_r}\) until it becomes row-standard in the sense \(\,i_1 \lt i_2 \lt \cdots i_r\). Hence, in the tensor product \[ \bigwedge^{\lambda_1} E \;\otimes\; \bigwedge^{\lambda_2} E \;\otimes\; \cdots \otimes \bigwedge^{\lambda_s} E, \] one sees that every tensor of wedge-products can be expressed as a linear combination of tensors in which each row (wedge) is written in ascending order. Accordingly, every element in \(L_\lambda E\) can be represented as a linear combination of row-standard tableaux. Therefore, once we pass to the quotient by the relations \(R(\lambda,E)\), all of those row-standard forms still span the resulting module \(L_\lambda E\). This is the intuitive reason why one says it is "clear" that \(\text{RST}(\lambda,[1,n])\) spans (generates) \(L_\lambda E\).

Example with shape \((2,2)\).

Step 0. Setup.

Step 1. A non-standard example. Consider the following \(2\times2\) tableau \[ T \;=\; \begin{array}{|c|c|} \hline 3 & 4 \\ \hline 1 & 2 \\ \hline \end{array}. \] The corresponding element of \(\;\wedge^2 E \otimes \wedge^2 E\;\) is \[ e_3 \wedge e_4 \;\;\otimes\;\; e_1 \wedge e_2. \] First, notice that each row (3,\,4) and (1,\,2) is strictly increasing, so row-standardness is satisfied. However, for the columns, we have \((3,1)\) in the first column and \((4,2)\) in the second column; neither \(3 \le 1\) nor \(4 \le 2\) holds! Hence \(T\) fails to be column-standard, so overall it is not a standard tableau.

Step 2. Applying a ``straightening relation.'' Theorem~6.1 (and Weyman's construction) shows that whenever \(T(1,w) > T(2,w)\) for some column \(w\), we can apply the submodule relations \(R_{1,2}(E)\) (coming from the maps \(\theta(\lambda,1,u,v;E)\)) to rewrite \(T\) modulo those relations as a linear combination of other tableaux, each of which is ``earlier'' in a certain order. By repeating such steps, one eventually expresses \([T]\) entirely in terms of standard tableaux. Concretely, in our example: \[ T(1,1)=3,\; T(2,1)=1 \quad\Longrightarrow\quad 3 > 1. \] We focus on the first column (``\(w=1\)''). The relevant map \(\theta\bigl((2,2),1,u,v;E\bigr)\) with \(u+v\lt 2\) forces a relation in the quotient, sometimes called a column-relation or straightening relation. When one carries out the resulting sum (which involves an ``exterior diagonal'' on \(e_3 \wedge e_4 \,\wedge\, e_1 \wedge e_2\)), the tableau \(\begin{smallmatrix}3 &4\\1 &2\end{smallmatrix}\) will appear with coefficient \(1\), plus a combination of new tableaux in which the smaller indices (like ``1'') have been ``promoted'' into the top row. Symbolically: \[ [\,e_3\wedge e_4 \;\otimes\; e_1\wedge e_2\,] \;=\; [T] \;=\; \bigl(\text{earlier tableaux}\bigr) \;-\; \bigl(\text{further shuffles}\bigr)\,. \] One then repeats on each of those new tableaux if they are still not column-standard.

Step 3. Outcome: a sum of standard tableaux. Eventually, this finite procedure (often called the ``straightening algorithm'') reaches a sum of fully standard tableaux. For instance, it is straightforward to check that \[ \begin{array}{|c|c|} \hline 1 & 2 \\ \hline 3 & 4 \\ \hline \end{array} \] is indeed standard (rows strictly increasing, columns weakly increasing). Hence, after applying the relations from \(R\left((2,2),E\right)\) plus the antisymmetry in each wedge, one finds \[ [T] \;=\; \left[\, e_3 \wedge e_4 \;\otimes\; e_1 \wedge e_2 \right] \;=\; \text{(some integer combination of) standard tableaux.} \] In particular, if you track the sign carefully, you find an explicit coefficient in front of the final standard tableau(s).

\textbf{Answer.} In the module $L_{(2,2)}E$ the non-standard tableau \[ T \;=\; \begin{array}{|c|c|} \hline 3 & 4 \\ \hline 1 & 2 \\ \hline \end{array} \] can be ``straightened'' uniquely to a linear combination of the two standard $2\times2$ tableaux with entries $\{1,2,3,4\}$. Those two standard tableaux are precisely \[ S_1 \;=\; \begin{array}{|c|c|} \hline 1 & 2 \\ \hline 3 & 4 \\ \hline \end{array} \quad\text{and}\quad S_2 \;=\; \begin{array}{|c|c|} \hline 1 & 3 \\ \hline 2 & 4 \\ \hline \end{array}. \] A careful tracking of the submodule relations $R_{1,2}(E)$, signs from wedge-antisymmetry, and the ``exterior diagonal'' shows that in the quotient $L_{(2,2)}E$ we have the identity \[ \bigl[\, e_3 \wedge e_4 \;\otimes\; e_1 \wedge e_2 \bigr] \;=\; \bigl[T\bigr] \;=\; \bigl[S_1\bigr] \;-\; \bigl[S_2\bigr]. \] Equivalently, in tableau notation: \[ \begin{array}{|c|c|} \hline 3 & 4 \\ \hline 1 & 2 \\ \hline \end{array} \;\;=\;\; \begin{array}{|c|c|} \hline 1 & 2 \\ \hline 3 & 4 \\ \hline \end{array} \;-\; \begin{array}{|c|c|} \hline 1 & 3 \\ \hline 2 & 4 \\ \hline \end{array} \;\;\text{in}\;L_{(2,2)}E. \] That is the final ``straightened'' expression of $T$ as a linear combination of standard tableaux.

Remark. If all one needed to fix was the row ordering (e.g.\ \((4,3)\) reorders to \(-\,(3,4)\) by wedge-antisymmetry), that does not require the Weyman relations: reordering a wedge is automatic from \(\,e_i \wedge e_j = -\,e_j \wedge e_i\). But fixing columns is precisely where the extra relations \(R\left((2,2),E\right)\) come into play. This is what ensures that column-standardness can be forced, eventually yielding a unique linear combination of genuinely standard tableaux as the final result.

Reference. "Cohomology of Vector Bundles and Syzygies". Page 36.

(6.2.) Theorem.

(a). \( L_{\lambda / \mu}E = \wedge^{\lambda_1 - \mu_1} E \otimes \wedge^{\lambda_2 - \mu_2} E \otimes \cdots \otimes \wedge^{\lambda_s - \mu_s} E / R(\lambda / \mu, E) \), where \( R(\lambda / \mu, E) \) is spanned by the subspaces:

(b). The standard tableaux of shape \( \lambda / \mu \) form a basis of \( L_{\lambda / \mu} \).

Reference. "Cohomology of Vector Bundles and Syzygies". Page 41.

References

• Weyman, Jerzy. Cohomology of Vector Bundles and Syzygies. Cambridge University Press, Cambridge, 2003.