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Statistic identifier: St000716
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Collection: Integer partitions
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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.
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References:
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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
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Created: Mar 21, 2017 at 08:32 by Martin Rubey
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Last Updated: Mar 21, 2017 at 08:32 by Martin Rubey