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> GetGeneralizedCobwebQuiver(int numVerticesInCenterPolygon, int numLayers, int firstVertex = DefaultFirstVertex)
{
if (!GeneralizedCobwebParameterIsValid(numVerticesInCenterPolygon, numLayers))
{
throw new ArgumentOutOfRangeException(nameof(numVerticesInCenterPolygon));
}
// Sort of backwards to construct the entire QP only to return just the quiver,
// but this reduces duplicated logic.
var qp = UsefulQPs.GetGeneralizedCobwebQP(numVerticesInCenterPolygon, numLayers, firstVertex);
return(qp.Quiver);
}
public static Quiver <int> GetPointedFlowerQuiver(int numPeriods, int firstVertex = DefaultFirstVertex)
{
if (!PointedFlowerParameterIsValid(numPeriods))
{
throw new ArgumentOutOfRangeException(nameof(numPeriods));
}
// Sort of backwards to construct the entire QP only to return just the quiver
// But this reduces duplicated logic
var qp = UsefulQPs.GetPointedFlowerQP(numPeriods, firstVertex);
return(qp.Quiver);
}
public static Quiver <int> GetEvenFlowerType2Quiver(int numVerticesInCenterPolygon, int firstVertex = DefaultFirstVertex)
{
if (!EvenFlowerType2ParameterIsValid(numVerticesInCenterPolygon))
{
throw new ArgumentOutOfRangeException(nameof(numVerticesInCenterPolygon));
}
// Sort of backwards to construct the entire QP only to return just the quiver
// But this reduces duplicated logic
var qp = UsefulQPs.GetEvenFlowerType2QP(numVerticesInCenterPolygon, firstVertex);
return(qp.Quiver);
}
public SelfInjectiveQP <int> GetSelfInjectiveCycleQP(int numVertices, int firstVertex = DefaultFirstVertex)
{
if (numVertices < 3)
{
throw new ArgumentOutOfRangeException(nameof(numVertices));
}
var qp = UsefulQPs.GetCycleQP(numVertices, firstVertex);
int n = numVertices;
var dict = Enumerable.Range(firstVertex, n).ToDictionary(k => k, k => (k - 2 - firstVertex).Modulo(n) + firstVertex);
var nakayamaPermutation = new NakayamaPermutation <int>(dict);
var selfInjectiveQp = new SelfInjectiveQP <int>(qp, nakayamaPermutation);
return(selfInjectiveQp);
}
/// <summary>
///
/// </summary>
/// <param name="numRows">The number of rows of vertices.</param>
/// <param name="firstVertex"></param>
/// <returns></returns>
public static Quiver <int> GetSquareQuiver(int numRows, int firstVertex = DefaultFirstVertex)
{
if (!SquareParameterIsValid(numRows))
{
throw new ArgumentOutOfRangeException(nameof(numRows));
}
if (numRows == 1)
{
return(new Quiver <int>(vertices: new int[] { firstVertex }, arrows: new Arrow <int> [0]));
}
// Sort of backwards to construct the entire QP only to return just the quiver
// But this reduces duplicated logic
var qp = UsefulQPs.GetSquareQP(numRows, firstVertex);
return(qp.Quiver);
}
public SelfInjectiveQP <int> GetSelfInjectiveSquareQP(int numRows, int firstVertex = DefaultFirstVertex)
{
if (!SquareParameterIsValid(numRows))
{
throw new ArgumentOutOfRangeException(nameof(numRows));
}
var qp = UsefulQPs.GetSquareQP(numRows);
int numVerticesInRow = numRows;
int numVertices = numRows * numVerticesInRow;
// Rotation "twice" clockwise/counterclockwise to get the Nakayama permutation
// This map happens to be just "x mapsto (n+1)-x" (labeling the vertices from 0 would be cleaner here I guess)
var nakayamaPermutation = Enumerable.Range(1, numVertices).ToDictionary(x => x, x => numVertices + 1 - x);
return(new SelfInjectiveQP <int>(qp, nakayamaPermutation));
}
/// <remarks>It is <em>assumed</em> that <em>all</em> odd flower QPs are self-injective.</remarks>
public SelfInjectiveQP <int> GetSelfInjectiveOddFlowerQP(int numVerticesInCenterPolygon, int firstVertex = DefaultFirstVertex)
{
if (numVerticesInCenterPolygon < 3 || numVerticesInCenterPolygon.Modulo(2) == 0)
{
throw new ArgumentOutOfRangeException(nameof(numVerticesInCenterPolygon));
}
var qp = UsefulQPs.GetOddFlowerQP(numVerticesInCenterPolygon);
int numLayers = (numVerticesInCenterPolygon + 1) / 2;
int numVerticesInFullInnerLayer = 2 * numVerticesInCenterPolygon;
int numVerticesInOuterLayer = 2 * numVerticesInFullInnerLayer;
// Rotate clockwise by 2*pi / ((numVerticesInCenterPolygon-1)/2) for the Nakayama permutation?
var nakayamaPermutation = new Dictionary <int, int>();
for (int layerIndex = 0; layerIndex < numLayers; layerIndex++)
{
var layer = GetLayerVertices(layerIndex);
for (int indexInLayer = 0; indexInLayer < layer.Count; indexInLayer++)
{
int input = layer[indexInLayer];
int stepsToRotateBy = (numVerticesInCenterPolygon - 1) / 2 * (layer.Count / numVerticesInCenterPolygon);
int output = layer[indexInLayer + stepsToRotateBy];
nakayamaPermutation[input] = output;
}
}
return(new SelfInjectiveQP <int>(qp, nakayamaPermutation));
CircularList <int> GetLayerVertices(int layerIndex) // 0-based layer index
{
if (layerIndex == 0)
{
return(new CircularList <int>(Enumerable.Range(1, numVerticesInCenterPolygon)));
}
int startVertex = layerIndex * numVerticesInFullInnerLayer - numVerticesInCenterPolygon + 1;
int numVertices = layerIndex < numLayers - 1 ? numVerticesInFullInnerLayer : numVerticesInOuterLayer;
return(new CircularList <int>(Enumerable.Range(startVertex, numVertices)));
}
}