***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information 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. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000716 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The dimension of the irreducible representation of Sp(6) labelled by an integer partition. Consider the symplectic group $Sp(2n)$. Then the integer partition $(\mu_1,\dots,\mu_k)$ of length at most $n$ corresponds to the weight vector $(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$. For example, the integer partition $(2)$ labels the symmetric square of the vector representation, whereas the integer partition $(1,1)$ labels the second fundamental representation. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(mu): C = CartanType("C3") if len(mu) <= C.rank() or (C.type()=="A" and len(mu) <= C.rank()+1): w = [m1-m2 for m1,m2 in zip(mu, mu[1:])] + [mu[-1]] + [0]*(C.rank()-len(mu)) return WeylDim(C, w) else: return 0 ----------------------------------------------------------------------------- Statistic values: [2] => 21 [1,1] => 14 [3] => 56 [2,1] => 64 [1,1,1] => 14 [4] => 126 [3,1] => 189 [2,2] => 90 [2,1,1] => 70 [1,1,1,1] => 0 [5] => 252 [4,1] => 448 [3,2] => 350 [3,1,1] => 216 [2,2,1] => 126 [2,1,1,1] => 0 [1,1,1,1,1] => 0 [6] => 462 [5,1] => 924 [4,2] => 924 [4,1,1] => 525 [3,3] => 385 [3,2,1] => 512 [3,1,1,1] => 0 [2,2,2] => 84 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 792 [6,1] => 1728 [5,2] => 2016 [5,1,1] => 1100 [4,3] => 1344 [4,2,1] => 1386 [4,1,1,1] => 0 [3,3,1] => 616 [3,2,2] => 378 [3,2,1,1] => 0 [3,1,1,1,1] => 0 [2,2,2,1] => 0 [2,2,1,1,1] => 0 [2,1,1,1,1,1] => 0 [1,1,1,1,1,1,1] => 0 [8] => 1287 [7,1] => 3003 [6,2] => 3900 [6,1,1] => 2079 [5,3] => 3276 [5,2,1] => 3072 [5,1,1,1] => 0 [4,4] => 1274 [4,3,1] => 2205 [4,2,2] => 1078 [4,2,1,1] => 0 [4,1,1,1,1] => 0 [3,3,2] => 594 [3,3,1,1] => 0 [3,2,2,1] => 0 [3,2,1,1,1] => 0 [3,1,1,1,1,1] => 0 [2,2,2,2] => 0 [2,2,2,1,1] => 0 [2,2,1,1,1,1] => 0 [2,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1] => 0 [9] => 2002 [8,1] => 4928 [7,2] => 6930 [7,1,1] => 3640 [6,3] => 6720 [6,2,1] => 6006 [6,1,1,1] => 0 [5,4] => 4116 [5,3,1] => 5460 [5,2,2] => 2464 [5,2,1,1] => 0 [5,1,1,1,1] => 0 [4,4,1] => 2184 [4,3,2] => 2240 [4,3,1,1] => 0 [4,2,2,1] => 0 [4,2,1,1,1] => 0 [4,1,1,1,1,1] => 0 [3,3,3] => 330 [3,3,2,1] => 0 [3,3,1,1,1] => 0 [3,2,2,2] => 0 [3,2,2,1,1] => 0 [3,2,1,1,1,1] => 0 [3,1,1,1,1,1,1] => 0 [2,2,2,2,1] => 0 [2,2,2,1,1,1] => 0 [2,2,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1] => 0 [10] => 3003 [9,1] => 7722 [8,2] => 11550 [8,1,1] => 6006 [7,3] => 12375 [7,2,1] => 10752 [7,1,1,1] => 0 [6,4] => 9450 [6,3,1] => 11319 [6,2,2] => 4914 [6,2,1,1] => 0 [6,1,1,1,1] => 0 [5,5] => 3528 [5,4,1] => 7168 [5,3,2] => 5720 [5,3,1,1] => 0 [5,2,2,1] => 0 [5,2,1,1,1] => 0 [5,1,1,1,1,1] => 0 [4,4,2] => 2457 [4,4,1,1] => 0 [4,3,3] => 1386 [4,3,2,1] => 0 [4,3,1,1,1] => 0 [4,2,2,2] => 0 [4,2,2,1,1] => 0 [4,2,1,1,1,1] => 0 [4,1,1,1,1,1,1] => 0 [3,3,3,1] => 0 [3,3,2,2] => 0 [3,3,2,1,1] => 0 [3,3,1,1,1,1] => 0 [3,2,2,2,1] => 0 [3,2,2,1,1,1] => 0 [3,2,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1] => 0 [2,2,2,2,2] => 0 [2,2,2,2,1,1] => 0 [2,2,2,1,1,1,1] => 0 [2,2,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1] => 0 [11] => 4368 [10,1] => 11648 [9,2] => 18304 [9,1,1] => 9450 [8,3] => 21120 [8,2,1] => 18018 [8,1,1,1] => 0 [7,4] => 18480 [7,3,1] => 21000 [7,2,2] => 8918 [7,2,1,1] => 0 [7,1,1,1,1] => 0 [6,5] => 10752 [6,4,1] => 16632 [6,3,2] => 12096 [6,3,1,1] => 0 [6,2,2,1] => 0 [6,2,1,1,1] => 0 [6,1,1,1,1,1] => 0 [5,5,1] => 6300 [5,4,2] => 8316 [5,4,1,1] => 0 [5,3,3] => 3744 [5,3,2,1] => 0 [5,3,1,1,1] => 0 [5,2,2,2] => 0 [5,2,2,1,1] => 0 [5,2,1,1,1,1] => 0 [5,1,1,1,1,1,1] => 0 [4,4,3] => 2002 [4,4,2,1] => 0 [4,4,1,1,1] => 0 [4,3,3,1] => 0 [4,3,2,2] => 0 [4,3,2,1,1] => 0 [4,3,1,1,1,1] => 0 [4,2,2,2,1] => 0 [4,2,2,1,1,1] => 0 [4,2,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1] => 0 [3,3,3,2] => 0 [3,3,3,1,1] => 0 [3,3,2,2,1] => 0 [3,3,2,1,1,1] => 0 [3,3,1,1,1,1,1] => 0 [3,2,2,2,2] => 0 [3,2,2,2,1,1] => 0 [3,2,2,1,1,1,1] => 0 [3,2,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,1] => 0 [2,2,2,2,1,1,1] => 0 [2,2,2,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1] => 0 [12] => 6188 [11,1] => 17017 [10,2] => 27846 [10,1,1] => 14300 [9,3] => 34034 [9,2,1] => 28672 [9,1,1,1] => 0 [8,4] => 32725 [8,3,1] => 36036 [8,2,2] => 15092 [8,2,1,1] => 0 [8,1,1,1,1] => 0 [7,5] => 23562 [7,4,1] => 32768 [7,3,2] => 22750 [7,3,1,1] => 0 [7,2,2,1] => 0 [7,2,1,1,1] => 0 [7,1,1,1,1,1] => 0 [6,6] => 8568 [6,5,1] => 19404 [6,4,2] => 19683 [6,4,1,1] => 0 [6,3,3] => 8190 [6,3,2,1] => 0 [6,3,1,1,1] => 0 [6,2,2,2] => 0 [6,2,2,1,1] => 0 [6,2,1,1,1,1] => 0 [6,1,1,1,1,1,1] => 0 [5,5,2] => 7700 [5,5,1,1] => 0 [5,4,3] => 7168 [5,4,2,1] => 0 [5,4,1,1,1] => 0 [5,3,3,1] => 0 [5,3,2,2] => 0 [5,3,2,1,1] => 0 [5,3,1,1,1,1] => 0 [5,2,2,2,1] => 0 [5,2,2,1,1,1] => 0 [5,2,1,1,1,1,1] => 0 [5,1,1,1,1,1,1,1] => 0 [4,4,4] => 1001 [4,4,3,1] => 0 [4,4,2,2] => 0 [4,4,2,1,1] => 0 [4,4,1,1,1,1] => 0 [4,3,3,2] => 0 [4,3,3,1,1] => 0 [4,3,2,2,1] => 0 [4,3,2,1,1,1] => 0 [4,3,1,1,1,1,1] => 0 [4,2,2,2,2] => 0 [4,2,2,2,1,1] => 0 [4,2,2,1,1,1,1] => 0 [4,2,1,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1,1] => 0 [3,3,3,3] => 0 [3,3,3,2,1] => 0 [3,3,3,1,1,1] => 0 [3,3,2,2,2] => 0 [3,3,2,2,1,1] => 0 [3,3,2,1,1,1,1] => 0 [3,3,1,1,1,1,1,1] => 0 [3,2,2,2,2,1] => 0 [3,2,2,2,1,1,1] => 0 [3,2,2,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,2] => 0 [2,2,2,2,2,1,1] => 0 [2,2,2,2,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: Mar 21, 2017 at 08:32 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Mar 21, 2017 at 08:32 by Martin Rubey