<Chapter Label="Combinatorial_Rep_Theory">
<Heading>Combinatorial representation theory</Heading>

<Section>
<Heading>Introduction</Heading> 

Here we introduce the implementation of the software package CREP
initially designed for MAPLE.
</Section>

<Section Label="UnitForms">
<Heading>Different unit forms</Heading>

<ManSection>
<Filt Type="Category" Name="IsUnitForm"/>
<Description>
The category for unit forms, which we identify with symmetric integral
matrices with 2 along the diagonal.
</Description>
</ManSection>

<ManSection>
  <Attr Name="BilinearFormOfUnitForm" Arg="B" />
  <Description>
    Arguments: <Arg>B</Arg> -- a unit form.<Br />
  </Description>
  <Returns>
    the bilinear form associated to a unit form <Arg>B</Arg>. 
  </Returns>
  <Description>The bilinear form associated to the unitform
  <Arg>B</Arg> given by a matrix <C>B</C> is defined for two vectors
  <C>x</C> and <C>y</C> as: <M>x*B*y^T</M>.
  </Description>
</ManSection>

<ManSection>
  <Prop Name="IsWeaklyNonnegativeUnitForm" Arg="B" />
  <Description>
    Arguments: <Arg>B</Arg> -- a unit form.<Br />
  </Description>
  <Returns>
    true is the unitform <Arg>B</Arg> is weakly non-negative, otherwise false. 
  </Returns>
  <Description>The unit form <Arg>B</Arg> is weakly non-negative is
  <M>B(x,y) \geq 0</M> for all <M>x\neq 0</M> in <M>\mathbb{Z}^n</M>, where
  <M>n</M> is the dimension of the square matrix associated to
  <Arg>B</Arg>.
  </Description>
</ManSection>

<ManSection>
  <Prop Name="IsWeaklyPositiveUnitForm" Arg="B" />
  <Description>
    Arguments: <Arg>B</Arg> -- a unit form.<Br />
  </Description>
  <Returns>
    true is the unitform <Arg>B</Arg> is weakly positive, otherwise false. 
  </Returns>
  <Description>The unit form <Arg>B</Arg> is weakly positive if
  <M>B(x,y) > 0</M> for all <M>x\neq 0</M> in <M>\mathbb{Z}^n</M>,
  where <M>n</M> is the dimension of the square matrix associated to
  <Arg>B</Arg>.
  </Description>
</ManSection>

<ManSection>
  <Attr Name="PositiveRootsOfUnitForm" Arg="B" />
  <Description>
    Arguments: <Arg>B</Arg> -- a unit form.<Br />
  </Description>
  <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!". 
  </Returns>
  <Description>
    This attribute will be attached to <Arg>B</Arg> when
    <C>IsWeaklyPositiveUnitForm</C> is applied to <Arg>B</Arg> and it
    is weakly positive.
  </Description>
</ManSection>


<ManSection>
  <Attr Name="QuadraticFormOfUnitForm" Arg="B" />
  <Description>
    Arguments: <Arg>B</Arg> -- a unit form.<Br />
  </Description>
  <Returns>
    the quadratic form associated to a unit form <Arg>B</Arg>. 
  </Returns>
  <Description>The quadratic form associated to the unitform
  <Arg>B</Arg> given by a matrix <C>B</C> is defined for a vector
  <C>x</C> as: <M>\frac{1}{2}x*B*x^T</M>.
  </Description>
</ManSection>

<ManSection>
  <Attr Name="SymmetricMatrixOfUnitForm" Arg="B" />
  <Description>
    Arguments: <Arg>B</Arg> -- a unit form.<Br />
  </Description>
  <Returns>
    the symmetric integral matrix which defines the unit form <Arg>B</Arg>. 
  </Returns>
</ManSection>

<ManSection>
  <Oper Name="TitsUnitFormOfAlgebra" Arg="A" />
  <Description>
    Arguments: <Arg>A</Arg> -- a finite dimensional (quotient of a)
    path algebra (by an admissible ideal).<Br />
  </Description>
  <Returns>
    the Tits unit form associated to the algebra <Arg>A</Arg>.
  </Returns>
  <Description>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 <M>B</M> such that for <M>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</M>, where <M>n</M> is the number of vertices
  in <M>Q</M>.
  </Description>
</ManSection>

<ManSection>
  <Oper Name="EulerBilinearFormOfAlgebra" Arg="A" />
  <Description>
    Arguments: <Arg>A</Arg> -- a finite dimensional (quotient of a)
    path algebra (by an admissible ideal).<Br />
  </Description>
  <Returns>
    the Euler (non-symmetric) bilinear form associated to the algebra <Arg>A</Arg>.
  </Returns>
  <Description>This function returns the Euler (non-symmetric) bilinear form 
  associated to a finite dimensional (basic) quotient of a path algebra <Arg>A</Arg>.
  That is,  it returns a bilinear form (function) defined by<Br />
  <M>f(x,y) = x*\textrm{CartanMatrix}(A)^{(-1)}*y</M><Br />
  It makes sense only in case <Arg>A</Arg> is of finite global dimension.
  </Description>
</ManSection>


<ManSection>
  <Oper Name="UnitForm" Arg="B" />
  <Description>
    Arguments: <Arg>B</Arg> -- an integral matrix.<Br />
  </Description>
  <Returns>
    the unit form in the category <Ref Filt="IsUnitForm"/> associated
    to the matrix <Arg>B</Arg>.
  </Returns>
  <Description>The function checks if <Arg>B</Arg> is a symmetric
  integral matrix with 2 along the diagonal, and returns an error
  message otherwise. In addition it sets the attributes, <Ref
  Attr="BilinearFormOfUnitForm"/>, <Ref
  Attr="QuadraticFormOfUnitForm"/> and <Ref
  Attr="SymmetricMatrixOfUnitForm"/>.
  </Description>
</ManSection>


</Section>
</Chapter>
