Weight Sets¶
Module for doing computations with the weights of a weight module and action of Weyl group.
-
class
bggcohomology.weight_set.WeightSet(root_system, hot_start=None)[source]¶ Class to do simple computations with the weights of a weight module.
Parameters: root_system (str) – String representing the root system (e.g. ‘A2’)
Variables: - root_system (str) – String representing the root system (e.g. ‘A2’)
- W (WeylGroup) – Object encoding the Weyl group.
- weyl_dic (dict(str, WeylGroup.element_class)) – Dictionary mapping strings representing a Weyl group element as reduced word in simple reflections, to the Weyl group element.
- reduced_words (list[str]) – Sorted list of all strings representing Weyl group elements as reduced word in simple reflections.
- simple_roots (list[RootSpace.element_class]) – List of the simple roots
- rank (int) – Rank of root system
- rho (RootSpace.element_class) – Half the sum of all positive roots
- pos_roots (List[RootSpace.element_class]) – List of all the positive roots
- action_dic (Dict[str, np.array(np.int32, np.int32)]) – dictionary mapping each string representing an element of the Weyl group to a matrix expressing the action on the simple roots.
- rho_action_dic (Dict[str, np.array(np.int32)]) – dictionary mapping each string representing an element of the Weyl group to a vector representing the image of the dot action on rho.
-
compute_weights(weights)[source]¶ Find dot-regular weights and associated dominant weights of a set of weights.
Parameters: weights (iterable(iterable(int))) – Iterable of weights Returns: list of triples consisting of dot-regular weight, associated dominant, and the length of the Weyl group element making the weight dominant under the dot action. Return type: list(tuple[tuple(int), tuple(int), int])
-
dot_action(w, mu)[source]¶ Compute the dot action of w on mu.
The dot action \(w\cdot\mu = w(\mu+\rho)-\rho\), with \(\rho\) half the sum of all the positive roots.
Parameters: - w (str) – string representing the weyl group element
- mu (iterable(int)) – the weight
Returns: vector encoding the new weight
Return type: np.array[np.int32]
-
dot_orbit(mu)[source]¶ Compute the orbit of the Weyl group action on a weight.
Parameters: mu (iterable(int)) – A weight Returns: Dictionary mapping Weyl group elements to weights encoded as numpy vectors. Return type: dict(str, np.array[np.int32])
-
classmethod
from_bgg(BGG)[source]¶ Initialize from an instance of BGGComplex.
Some data can be reused, and this gives roughly 3x faster initialization.
Parameters: BGG (BGGComplex) – The BGGComplex to initialize from.
-
get_action_dic()[source]¶ Compute weyl group action as well as action on rho.
Returns: - Dict[str, np.array(np.int32, np.int32)] (dictionary mapping each string representing an) – element of the Weyl group to a matrix expressing the action on the simple roots.
- Dict[str, np.array(np.int32)] (dictionary mapping each string representing an) – element of the Weyl group to a vector representing the image of the dot action on rho.
-
get_vertex_weights(mu)[source]¶ For a given dot-regular mu, return its orbit under the dot-action.
Parameters: mu (iterable(int)) – Returns: list of weights Return type: list[tuple[int]]
-
highest_weight_rep_dim(mu)[source]¶ Give dimension of highest weight representation of integral dominant weight.
Parameters: mu (tuple(int)) – A integral dominant weight Returns: dimension of highest weight representation. Return type: int
-
is_dominant(mu)[source]¶ Use sagemath built-in function to check if weight is dominant.
Parameters: mu (iterable(int)) – the weight Returns: True if weight is dominant Return type: bool
-
is_dot_regular(mu)[source]¶ Check if mu has a non-trivial stabilizer under the dot action.
Parameters: mu (iterable(int)) – The weight Returns: True if the weight is dot-regular Return type: bool
-
make_dominant(mu)[source]¶ For a dot-regular weight mu, w such that if w.mu is dominant.
Such a w exists iff mu is dot-regular, in which case it is also unique.Parameters: mu (iterable(int)) – the dot-regular weight Returns: - tuple(int) – The dominant weight w.mu
- str – the string representing the Weyl group element w.