Index 4

Partitions

In this case there is only one possible partition of 48.

Mathematica Notebook

Mathematica Notebook that determines the possible isotropy weights

Results

List of the possible isotropy weights and multigraphs for the index-4 case.

Partitions

In this case there is only one possible partition of 48: the one with all 12 elements equal to 4.

Mathematica Notebook

This Mathematica notebook determines the possible isotropy weights for circle actions on a 8-dim'l manifold of index 4 with 6 fixed points that extends to a 2-torus action. It is divided in several parts described below.

Part 3

Determines which matrices are singular for the only possible partition, dividing them into 2 sets according to their ranks.

Part 4

Considers the first type of matrices producing a list of the corresponding isotropy weights, checking if they satisfy the polynomial equations in (1.1) of [GS].

Part 5

Considers the second type of matrices:
a) Selects those whose null spaces contain vectors with positive entries.
b) Produces the list of the corresponding isotropy weights that satisfy the polynomial equations in (1.1) of [GS].
c) Selects those isotropy weights that at a fixed point of index 2i the first i weights are negative and the others are positive.

At each step, the resulting lists of isotropy weights are saved in different files so that it is easy to verify which isotropy weights are discarded with each test performed.

Part 6

Lists all possible isotropy weights.

Results

List of the possible isotropy weights and multigraphs for the index-4 case.

© 2024 Leonor Godinho, Nicholas Lindsay and Silvia Sabatini

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