public SelfInjectiveQP <int> GetSelfInjectiveTriangleQP(int numRows, int firstVertex = DefaultFirstVertex)
{
if (numRows < 2)
{
throw new ArgumentOutOfRangeException(nameof(numRows));
}
var qp = UsefulQPs.GetTriangleQP(numRows);
var potential = new Potential <int>();
// Rotation "once" clockwise to get the Nakayama permutation
var nakayamaPermutation = new Dictionary <int, int>();
// Start with the right-most "column" (1, 3, 6, 10, ...), which is mapped to the bottom row,
// and go left to (2, 5, 9, ...), which is mapped to the second last row, and so on.
var columnVertices = Enumerable.Range(1, numRows).Select(k => Utility.TriangularNumber(k));
for (int rowIndex = numRows; rowIndex >= 1; rowIndex--)
{
var rowVertices = Enumerable.Range(Utility.TriangularNumber(rowIndex - 1) + 1, rowIndex).Reverse();
var inputOutputPairs = columnVertices.Zip(rowVertices, (x, y) => (x, y));
foreach (var(x, y) in inputOutputPairs)
{
nakayamaPermutation[x] = y;
}
columnVertices = columnVertices.Select(x => x - 1).Skip(1);
}
return(new SelfInjectiveQP <int>(qp, nakayamaPermutation));
}
public static Quiver <int> GetTriangleQuiver(int numRows, int firstVertex = DefaultFirstVertex)
{
if (!TriangleParameterIsValid(numRows))
{
throw new ArgumentOutOfRangeException(nameof(numRows));
}
if (numRows == 1)
{
var vertices = new int[] { firstVertex };
var arrows = new Arrow <int>[] { };
return(new Quiver <int>(vertices, arrows));
}
// Sort of backwards to construct the entire QP only to return just the quiver
// But this reduces duplicated logic
var qp = UsefulQPs.GetTriangleQP(numRows, firstVertex);
return(qp.Quiver);
}