GAP (QPA) - Chapter 12: Combinatorial representation theory

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12 Combinatorial representation theory
12.1 Introduction
12.2 Different unit forms
12.2-1 IsUnitForm
12.2-2 BilinearFormOfUnitForm
12.2-3 IsWeaklyNonnegativeUnitForm
12.2-4 IsWeaklyPositiveUnitForm
12.2-5 PositiveRootsOfUnitForm
12.2-6 QuadraticFormOfUnitForm
12.2-7 SymmetricMatrixOfUnitForm
12.2-8 TitsUnitFormOfAlgebra
12.2-9 EulerBilinearFormOfAlgebra
12.2-10 UnitForm

### 12 Combinatorial representation theory

#### 12.1 Introduction

Here we introduce the implementation of the software package CREP initially designed for MAPLE.

#### 12.2 Different unit forms

##### 12.2-1 IsUnitForm

`‣ IsUnitForm` | ( category )

The category for unit forms, which we identify with symmetric integral matrices with 2 along the diagonal.

##### 12.2-2 BilinearFormOfUnitForm

`‣ BilinearFormOfUnitForm`( B ) | ( attribute )

Arguments: B -- a unit form.

Returns: the bilinear form associated to a unit form B.

The bilinear form associated to the unitform B given by a matrix `B` is defined for two vectors `x` and `y` as: \(x*B*y^T\).

##### 12.2-3 IsWeaklyNonnegativeUnitForm

`‣ IsWeaklyNonnegativeUnitForm`( B ) | ( property )

Arguments: B -- a unit form.

Returns: true is the unitform B is weakly non-negative, otherwise false.

The unit form B is weakly non-negative is \(B(x,y) \geq 0\) for all \(x\neq 0\) in \(\mathbb{Z}^n\), where \(n\) is the dimension of the square matrix associated to B.

##### 12.2-4 IsWeaklyPositiveUnitForm

`‣ IsWeaklyPositiveUnitForm`( B ) | ( property )

Arguments: B -- a unit form.

Returns: true is the unitform B is weakly positive, otherwise false.

The unit form B is weakly positive if \(B(x,y) > 0\) for all \(x\neq 0\) in \(\mathbb{Z}^n\), where \(n\) is the dimension of the square matrix associated to B.

##### 12.2-5 PositiveRootsOfUnitForm

`‣ PositiveRootsOfUnitForm`( B ) | ( attribute )

Arguments: B -- a unit form.

Returns: the positive roots of a unit form, if the unit form is weakly positive. If they have not been computed, an error message will be returned saying "no method found!".

This attribute will be attached to B when `IsWeaklyPositiveUnitForm` is applied to B and it is weakly positive.

##### 12.2-6 QuadraticFormOfUnitForm

`‣ QuadraticFormOfUnitForm`( B ) | ( attribute )

Arguments: B -- a unit form.

Returns: the quadratic form associated to a unit form B.

The quadratic form associated to the unitform B given by a matrix `B` is defined for a vector `x` as: \(\frac{1}{2}x*B*x^T\).

##### 12.2-7 SymmetricMatrixOfUnitForm

`‣ SymmetricMatrixOfUnitForm`( B ) | ( attribute )

Arguments: B -- a unit form.

Returns: the symmetric integral matrix which defines the unit form B.

##### 12.2-8 TitsUnitFormOfAlgebra

`‣ TitsUnitFormOfAlgebra`( A ) | ( operation )

Arguments: A -- a finite dimensional (quotient of a) path algebra (by an admissible ideal).

Returns: the Tits unit form associated to the algebra A.

This function returns the Tits unitform associated to a finite dimensional quotient of a path algebra by an admissible ideal or path algebra, given that the underlying quiver has no loops or minimal relations that starts and ends in the same vertex. That is, then it returns a symmetric matrix \(B\) such that for \(x = (x_1,...,x_n) (1/2)*(x_1,...,x_n)B(x_1,...,x_n)^T = \sum_{i=1}^n x_i^2 - \sum_{i,j} \dim_k \Ext^1(S_i,S_j)x_ix_j + \sum_{i,j} \dim_k \Ext^2(S_i,S_j)x_ix_j\), where \(n\) is the number of vertices in \(Q\).

##### 12.2-9 EulerBilinearFormOfAlgebra

`‣ EulerBilinearFormOfAlgebra`( A ) | ( operation )

Arguments: A -- a finite dimensional (quotient of a) path algebra (by an admissible ideal).

Returns: the Euler (non-symmetric) bilinear form associated to the algebra A.

This function returns the Euler (non-symmetric) bilinear form associated to a finite dimensional (basic) quotient of a path algebra A. That is, it returns a bilinear form (function) defined by
\(f(x,y) = x*\textrm{CartanMatrix}(A)^{(-1)}*y\)
It makes sense only in case A is of finite global dimension.

##### 12.2-10 UnitForm

`‣ UnitForm`( B ) | ( operation )

Arguments: B -- an integral matrix.

Returns: the unit form in the category `IsUnitForm` (12.2-1) associated to the matrix B.

The function checks if B is a symmetric integral matrix with 2 along the diagonal, and returns an error message otherwise. In addition it sets the attributes, `BilinearFormOfUnitForm` (12.2-2), `QuadraticFormOfUnitForm` (12.2-6) and `SymmetricMatrixOfUnitForm` (12.2-7).

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