Coverage for local/lib/python2.7/site-packages/sage/combinat/rigged_configurations/bij_type_D_twisted.py : 74%

r""" 

Bijection classes for type `D_{n+1}^{(2)}`. 

Part of the (internal) classes which runs the bijection between rigged 

configurations and KR tableaux of type `D_{n+1}^{(2)}`. 

AUTHORS: 

- Travis Scrimshaw (2011-04-15): Initial version 

TESTS:: 

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 2], [[2,1]]) 

sage: from sage.combinat.rigged_configurations.bij_type_D_twisted import KRTToRCBijectionTypeDTwisted 

sage: bijection = KRTToRCBijectionTypeDTwisted(KRT(pathlist=[[-1,2]])) 

sage: TestSuite(bijection).run() 

sage: RC = RiggedConfigurations(['D', 4, 2], [[2, 1]]) 

sage: from sage.combinat.rigged_configurations.bij_type_D_twisted import RCToKRTBijectionTypeDTwisted 

sage: bijection = RCToKRTBijectionTypeDTwisted(RC()) 

sage: TestSuite(bijection).run() 

""" 

#***************************************************************************** 

# Copyright (C) 2011, 2012 Travis Scrimshaw <tscrim@ucdavis.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.combinat.rigged_configurations.bij_type_A import KRTToRCBijectionTypeA 

from sage.combinat.rigged_configurations.bij_type_A2_even import KRTToRCBijectionTypeA2Even 

from sage.combinat.rigged_configurations.bij_type_A2_even import RCToKRTBijectionTypeA2Even 

from sage.combinat.rigged_configurations.bij_type_D import KRTToRCBijectionTypeD 

from sage.combinat.rigged_configurations.bij_type_D import RCToKRTBijectionTypeD 

class KRTToRCBijectionTypeDTwisted(KRTToRCBijectionTypeD, KRTToRCBijectionTypeA2Even): 

r""" 

Specific implementation of the bijection from KR tableaux to rigged 

configurations for type `D_{n+1}^{(2)}`. 

This inherits from type `C_n^{(1)}` and `D_n^{(1)}` because we use the 

same methods in some places. 

""" 

def run(self, verbose=False): 

""" 

Run the bijection from a tensor product of KR tableaux to a rigged 

configuration for type `D_{n+1}^{(2)}`. 

INPUT: 

- ``tp_krt`` -- A tensor product of KR tableaux 

- ``verbose`` -- (Default: ``False``) Display each step in the 

bijection 

EXAMPLES:: 

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 2], [[3,1]]) 

sage: from sage.combinat.rigged_configurations.bij_type_D_twisted import KRTToRCBijectionTypeDTwisted 

sage: KRTToRCBijectionTypeDTwisted(KRT(pathlist=[[-1,3,2]])).run() 

<BLANKLINE> 

-1[ ]-1 

<BLANKLINE> 

0[ ]0 

<BLANKLINE> 

1[ ]1 

<BLANKLINE> 

""" 

if verbose: 

from sage.combinat.rigged_configurations.tensor_product_kr_tableaux_element \ 

import TensorProductOfKirillovReshetikhinTableauxElement 

for cur_crystal in reversed(self.tp_krt): 

r = cur_crystal.parent().r() 

# Iterate through the columns 

for col_number, cur_column in enumerate(reversed(cur_crystal.to_array(False))): 

self.cur_path.insert(0, []) # Prepend an empty list 

# Check to see if we are a spinor column 

if r == self.n: 

if verbose: 

print("====================") 

print(repr(TensorProductOfKirillovReshetikhinTableauxElement(self.tp_krt.parent(), self.cur_path))) 

print("--------------------") 

print(repr(self.ret_rig_con)) 

print("--------------------\n") 

print("Applying doubling map") 

self.doubling_map() 

self.cur_dims.insert(0, [0, 1]) 

for letter in reversed(cur_column): 

self.cur_dims[0][0] += 1 

val = letter.value # Convert from a CrystalOfLetter to an Integer 

if verbose: 

print("====================") 

print(repr(TensorProductOfKirillovReshetikhinTableauxElement(self.tp_krt.parent(), self.cur_path))) 

print("--------------------") 

print(repr(self.ret_rig_con)) 

print("--------------------\n") 

# Build the next state 

self.cur_path[0].insert(0, [letter]) # Prepend the value 

self.next_state(val) 

# Check to see if we are a spinor column 

if r == self.n: 

if verbose: 

print("====================") 

print(repr(TensorProductOfKirillovReshetikhinTableauxElement(self.tp_krt.parent(), self.cur_path))) 

print("--------------------") 

print(repr(self.ret_rig_con)) 

print("--------------------\n") 

print("Applying halving map") 

self.halving_map() 

# If we've split off a column, we need to merge the current column 

# to the current crystal tableau 

if col_number > 0: 

for i, letter_singleton in enumerate(self.cur_path[0]): 

self.cur_path[1][i].insert(0, letter_singleton[0]) 

self.cur_dims[1][1] += 1 

self.cur_path.pop(0) 

self.cur_dims.pop(0) 

# And perform the inverse column splitting map on the RC 

for a in range(self.n): 

self._update_vacancy_nums(a) 

self.ret_rig_con.set_immutable() # Return it to immutable 

return self.ret_rig_con 

def next_state(self, val): 

r""" 

Build the next state for type `D_{n+1}^{(2)}`. 

TESTS:: 

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 2], [[2,1]]) 

sage: from sage.combinat.rigged_configurations.bij_type_D_twisted import KRTToRCBijectionTypeDTwisted 

sage: bijection = KRTToRCBijectionTypeDTwisted(KRT(pathlist=[[-1,2]])) 

sage: bijection.cur_path.insert(0, []) 

sage: bijection.cur_dims.insert(0, [0, 1]) 

sage: bijection.cur_path[0].insert(0, [2]) 

sage: bijection.next_state(2) 

""" 

n = self.n 

tableau_height = len(self.cur_path[0]) - 1 

if val == 'E': 

KRTToRCBijectionTypeA2Even.next_state(self, val) 

return 

elif val > 0: 

# If it is a regular value, we follow the A_n rules 

KRTToRCBijectionTypeA.next_state(self, val) 

if tableau_height >= n - 1: 

self._correct_vacancy_nums() 

return 

pos_val = -val 

if pos_val == 0: 

if len(self.ret_rig_con[pos_val - 1]) > 0: 

max_width = self.ret_rig_con[n-1][0] 

else: 

max_width = 1 

max_width = self.ret_rig_con[n-1].insert_cell(max_width) 

width_n = max_width + 1 

# Follow regular A_n rules 

for a in reversed(range(tableau_height, n-1)): 

max_width = self.ret_rig_con[a].insert_cell(max_width) 

self._update_vacancy_nums(a + 1) 

self._update_partition_values(a + 1) 

self._update_vacancy_nums(tableau_height) 

if tableau_height >= n - 1: 

self._correct_vacancy_nums() 

self._update_partition_values(tableau_height) 

if tableau_height > 0: 

self._update_vacancy_nums(tableau_height-1) 

self._update_partition_values(tableau_height-1) 

# Make the new string at n quasi-singular 

p = self.ret_rig_con[n-1] 

num_rows = len(p) 

for i in range(num_rows): 

if p._list[i] == width_n: 

j = i+1 

while j < num_rows and p._list[j] == width_n \ 

and p.vacancy_numbers[j] == p.rigging[j]: 

j += 1 

p.rigging[j-1] -= 1 

break 

return 

case_S = [None] * n 

pos_val = -val 

# Always add a cell to the first singular value in the first 

# tableau we are updating. 

if len(self.ret_rig_con[pos_val - 1]) > 0: 

max_width = self.ret_rig_con[pos_val - 1][0] 

else: 

max_width = 1 

# Add cells similar to type A_n but we move to the right until we 

# reach the value of n-1 

for a in range(pos_val - 1, n-1): 

max_width = self.ret_rig_con[a].insert_cell(max_width) 

case_S[a] = max_width 

# Special case for n 

# If we find a quasi-singular string first, then we are in case (Q, S) 

# otherwise we will find a singular string and insert 2 cells 

partition = self.ret_rig_con[n-1] 

num_rows = len(partition) 

case_QS = False 

for i in range(num_rows + 1): 

if i == num_rows: 

max_width = 0 

if case_QS: 

partition._list.append(1) 

partition.vacancy_numbers.append(None) 

# Go through our partition until we find a length of greater than 1 

j = len(partition._list) - 1 

while j >= 0 and partition._list[j] == 1: 

j -= 1 

partition.rigging.insert(j + 1, None) 

width_n = 1 

else: 

# Go through our partition until we find a length of greater than 2 

j = len(partition._list) - 1 

while j >= 0 and partition._list[j] <= 2: 

j -= 1 

partition._list.insert(j+1, 2) 

partition.vacancy_numbers.insert(j+1, None) 

partition.rigging.insert(j+1, None) 

break 

elif partition._list[i] <= max_width: 

if partition.vacancy_numbers[i] == partition.rigging[i]: 

max_width = partition._list[i] 

if case_QS: 

partition._list[i] += 1 

width_n = partition._list[i] 

partition.rigging[i] = None 

else: 

j = i - 1 

while j >= 0 and partition._list[j] <= max_width + 2: 

partition.rigging[j+1] = partition.rigging[j] # Shuffle it along 

j -= 1 

partition._list.pop(i) 

partition._list.insert(j+1, max_width + 2) 

partition.rigging[j+1] = None 

break 

elif partition.vacancy_numbers[i] - 1 == partition.rigging[i] and not case_QS: 

case_QS = True 

partition._list[i] += 1 

partition.rigging[i] = None 

# No need to set max_width here since we will find a singular string 

# Now go back following the regular C_n (ish) rules 

for a in reversed(range(tableau_height, n-1)): 

if case_S[a] == max_width: 

self._insert_cell_case_S(self.ret_rig_con[a]) 

else: 

max_width = self.ret_rig_con[a].insert_cell(max_width) 

self._update_vacancy_nums(a + 1) 

self._update_partition_values(a + 1) 

# Update the final rigged partitions 

self._update_vacancy_nums(tableau_height) 

if tableau_height >= n - 1: 

self._correct_vacancy_nums() 

self._update_partition_values(tableau_height) 

if pos_val <= tableau_height: 

for a in range(pos_val-1, tableau_height): 

self._update_vacancy_nums(a) 

self._update_partition_values(a) 

if pos_val > 1: 

self._update_vacancy_nums(pos_val - 2) 

self._update_partition_values(pos_val - 2) 

elif tableau_height > 0: 

self._update_vacancy_nums(tableau_height - 1) 

self._update_partition_values(tableau_height - 1) 

if case_QS: 

# Make the new string quasi-singular 

num_rows = len(partition) 

for i in range(num_rows): 

if partition._list[i] == width_n: 

j = i+1 

while j < num_rows and partition._list[j] == width_n \ 

and partition.vacancy_numbers[j] == partition.rigging[j]: 

j += 1 

partition.rigging[j-1] -= 1 

break 

class RCToKRTBijectionTypeDTwisted(RCToKRTBijectionTypeD, RCToKRTBijectionTypeA2Even): 

r""" 

Specific implementation of the bijection from rigged configurations to 

tensor products of KR tableaux for type `D_{n+1}^{(2)}`. 

""" 

def run(self, verbose=False, build_graph=False): 

""" 

Run the bijection from rigged configurations to tensor product of KR 

tableaux for type `D_{n+1}^{(2)}`. 

INPUT: 

- ``verbose`` -- (default: ``False``) display each step in the 

bijection 

- ``build_graph`` -- (default: ``False``) build the graph of each 

step of the bijection 

EXAMPLES:: 

sage: RC = RiggedConfigurations(['D', 4, 2], [[3, 1]]) 

sage: x = RC(partition_list=[[],[1],[1]]) 

sage: from sage.combinat.rigged_configurations.bij_type_D_twisted import RCToKRTBijectionTypeDTwisted 

sage: RCToKRTBijectionTypeDTwisted(x).run() 

[[1], [3], [-2]] 

sage: bij = RCToKRTBijectionTypeDTwisted(x) 

sage: bij.run(build_graph=True) 

[[1], [3], [-2]] 

sage: bij._graph 

Digraph on 6 vertices 

""" 

from sage.combinat.crystals.letters import CrystalOfLetters 

letters = CrystalOfLetters(self.rigged_con.parent()._cartan_type.classical()) 

# This is technically bad, but because the first thing we do is append 

# an empty list to ret_crystal_path, we correct this. We do it this 

# way so that we do not have to remove an empty list after the 

# bijection has been performed. 

ret_crystal_path = [] 

for dim in self.rigged_con.parent().dims: 

ret_crystal_path.append([]) 

# Iterate over each column 

for dummy_var in range(dim[1]): 

# Split off a new column if necessary 

if self.cur_dims[0][1] > 1: 

self.cur_dims[0][1] -= 1 

self.cur_dims.insert(0, [dim[0], 1]) 

# Perform the corresponding splitting map on rigged configurations 

# All it does is update the vacancy numbers on the RC side 

for a in range(self.n): 

self._update_vacancy_numbers(a) 

if build_graph: 

y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions], use_vacancy_numbers=True) 

self._graph.append([self._graph[-1][1], (y, len(self._graph)), 'ls']) 

# Check to see if we are a spinor 

if dim[0] == self.n: 

if verbose: 

print("====================") 

print(repr(self.rigged_con.parent()(*self.cur_partitions, use_vacancy_numbers=True))) 

print("--------------------") 

print(ret_crystal_path) 

print("--------------------\n") 

print("Applying doubling map") 

self.doubling_map() 

if build_graph: 

y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions], use_vacancy_numbers=True) 

self._graph.append([self._graph[-1][1], (y, len(self._graph)), '2x']) 

while self.cur_dims[0][0] > 0: 

if verbose: 

print("====================") 

print(repr(self.rigged_con.parent()(*self.cur_partitions, use_vacancy_numbers=True))) 

print("--------------------") 

print(ret_crystal_path) 

print("--------------------\n") 

self.cur_dims[0][0] -= 1 # This takes care of the indexing 

b = self.next_state(self.cur_dims[0][0]) 

# Make sure we have a crystal letter 

ret_crystal_path[-1].append(letters(b)) # Append the rank 

if build_graph: 

y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions], use_vacancy_numbers=True) 

self._graph.append([self._graph[-1][1], (y, len(self._graph)), letters(b)]) 

self.cur_dims.pop(0) # Pop off the leading column 

# Check to see if we were a spinor 

if dim[0] == self.n: 

if verbose: 

print("====================") 

print(repr(self.rigged_con.parent()(*self.cur_partitions))) 

print("--------------------") 

print(ret_crystal_path) 

print("--------------------\n") 

print("Applying halving map") 

self.halving_map() 

if build_graph: 

y = self.rigged_con.parent()(*[x._clone() for x in self.cur_partitions], use_vacancy_numbers=True) 

self._graph.append([self._graph[-1][1], (y, len(self._graph)), '1/2x']) 

if build_graph: 

self._graph.pop(0) # Remove the dummy at the start 

from sage.graphs.digraph import DiGraph 

from sage.graphs.dot2tex_utils import have_dot2tex 

self._graph = DiGraph(self._graph) 

if have_dot2tex(): 

self._graph.set_latex_options(format="dot2tex", edge_labels=True) 

return self.KRT(pathlist=ret_crystal_path) 

def next_state(self, height): 

r""" 

Build the next state for type `D_{n+1}^{(2)}`. 

TESTS:: 

sage: RC = RiggedConfigurations(['D', 4, 2], [[2, 1]]) 

sage: from sage.combinat.rigged_configurations.bij_type_D_twisted import RCToKRTBijectionTypeDTwisted 

sage: bijection = RCToKRTBijectionTypeDTwisted(RC(partition_list=[[2],[2,2],[2,2]])) 

sage: bijection.next_state(0) 

-1 

""" 

n = self.n 

ell = [None] * (2*n) 

case_S = [False] * n 

case_Q = False 

b = None 

# Calculate the rank and ell values 

last_size = 0 

for a in range(height, n-1): 

ell[a] = self._find_singular_string(self.cur_partitions[a], last_size) 

if ell[a] is None: 

b = a + 1 

break 

else: 

last_size = self.cur_partitions[a][ell[a]] 

if b is None: 

partition = self.cur_partitions[n-1] 

# Modified version of _find_singular_string() 

for i in reversed(range(len(partition))): 

if partition[i] >= last_size: 

if partition.vacancy_numbers[i] == partition.rigging[i]: 

if partition[i] == 1: 

b = 'E' 

else: 

last_size = partition[i] 

case_S[n-1] = True 

ell[2*n-1] = i 

break 

elif partition.vacancy_numbers[i] - 1 == partition.rigging[i] and not case_Q: 

case_Q = True 

# Check if it is singular as well 

block_size = partition[i] 

for j in reversed(range(i)): 

if partition[j] != block_size: 

break 

elif partition.vacancy_numbers[j] == partition.rigging[j]: 

case_Q = False 

break 

if case_Q: 

last_size = partition[i] + 1 

ell[n-1] = i 

if ell[2*n-1] is None: 

if not case_Q: 

b = n 

else: 

b = 0 

if b is None: 

# Now go back 

for a in reversed(range(n-1)): 

if a >= height and self.cur_partitions[a][ell[a]] == last_size: 

ell[n+a] = ell[a] 

case_S[a] = True 

else: 

ell[n+a] = self._find_singular_string(self.cur_partitions[a], last_size) 

if ell[n + a] is None: 

b = -(a + 2) 

break 

else: 

last_size = self.cur_partitions[a][ell[n + a]] 

if b is None: 

b = -1 

# Determine the new rigged configuration by removing boxes from the 

# selected string and then making the new string singular 

if case_S[0]: 

row_num = None 

row_num_bar = self.cur_partitions[0].remove_cell(ell[n], 2) 

else: 

row_num = self.cur_partitions[0].remove_cell(ell[0]) 

row_num_bar = self.cur_partitions[0].remove_cell(ell[n]) 

for a in range(1, n-1): 

if case_S[a]: 

row_num_next = None 

row_num_bar_next = self.cur_partitions[a].remove_cell(ell[n+a], 2) 

else: 

row_num_next = self.cur_partitions[a].remove_cell(ell[a]) 

row_num_bar_next = self.cur_partitions[a].remove_cell(ell[n+a]) 

self._update_vacancy_numbers(a - 1) 

if row_num is not None: 

self.cur_partitions[a-1].rigging[row_num] = self.cur_partitions[a-1].vacancy_numbers[row_num] 

if row_num_bar is not None: 

self.cur_partitions[a-1].rigging[row_num_bar] = self.cur_partitions[a-1].vacancy_numbers[row_num_bar] 

row_num = row_num_next 

row_num_bar = row_num_bar_next 

if case_Q: 

row_num_next = self.cur_partitions[n-1].remove_cell(ell[n-1]) 

if case_S[n-1]: 

row_num_bar_next = self.cur_partitions[n-1].remove_cell(ell[2*n-1]) 

else: 

row_num_bar_next = None 

elif case_S[n-1]: 

row_num_next = None 

row_num_bar_next = self.cur_partitions[n-1].remove_cell(ell[2*n-1], 2) 

else: 

row_num_next = None 

row_num_bar_next = None 

self._update_vacancy_numbers(n - 2) 

if row_num is not None: 

self.cur_partitions[n-2].rigging[row_num] = self.cur_partitions[n-2].vacancy_numbers[row_num] 

if row_num_bar is not None: 

self.cur_partitions[n-2].rigging[row_num_bar] = self.cur_partitions[n-2].vacancy_numbers[row_num_bar] 

self._update_vacancy_numbers(n - 1) 

if height == n: 

self._correct_vacancy_nums() 

if row_num_next is not None: 

self.cur_partitions[n-1].rigging[row_num_next] = self.cur_partitions[n-1].vacancy_numbers[row_num_next] 

if row_num_bar_next is not None: 

if case_Q: 

vac_num = self.cur_partitions[n-1].vacancy_numbers[row_num_bar_next] 

self.cur_partitions[n-1].rigging[row_num_bar_next] = vac_num 

block_len = self.cur_partitions[n-1][row_num_bar_next] 

j = row_num_bar_next + 1 

length = len(self.cur_partitions[n-1]) 

# Find the place for the quasisingular rigging 

while j < length and self.cur_partitions[n-1][j] == block_len \ 

and self.cur_partitions[n-1].rigging[j] == vac_num: 

j += 1 

self.cur_partitions[n-1].rigging[j-1] = vac_num - 1 

else: 

self.cur_partitions[n-1].rigging[row_num_bar_next] = self.cur_partitions[n-1].vacancy_numbers[row_num_bar_next] 

return(b)