**Identifier**

Identifier

Values

['A',1]
=>
3

['A',2]
=>
8

['B',2]
=>
5

['G',2]
=>
7

['A',3]
=>
15

['B',3]
=>
7

['C',3]
=>
14

['A',4]
=>
24

['B',4]
=>
9

['C',4]
=>
27

['D',4]
=>
28

['F',4]
=>
26

['A',5]
=>
35

['B',5]
=>
11

['C',5]
=>
44

['D',5]
=>
45

['A',6]
=>
48

['B',6]
=>
13

['C',6]
=>
65

['D',6]
=>
66

['E',6]
=>
78

['A',7]
=>
63

['B',7]
=>
15

['C',7]
=>
90

['D',7]
=>
91

['E',7]
=>
133

['A',8]
=>
80

['B',8]
=>
17

['C',8]
=>
119

['D',8]
=>
120

['E',8]
=>
248

Description

The dimension of the quasi-minuscule representation of the Lie group of given type.

For every simple type there is a unique quasi-minuscule representation, and the unique dominant short root is its highest weight, see [2].

For every simple type there is a unique quasi-minuscule representation, and the unique dominant short root is its highest weight, see [2].

References

[1] wikipedia:Minuscule representation

[2]

[2]

**van Leeuwen, M.***quasi-minuscule representations*MathOverflow:129985Code

def statistic(C): n = C.rank() T = C.type() if T == "A": return n^2+2*n # adjoint if T == "B": return 2*n+1 # vector if T == "C": return 2*n^2-n-1 if T == "D": return 2*n^2-n # adjoint if T == "E": if n == 6: return 78 # adjoint if n == 7: return 133 # adjoint if n == 8: return 248 # adjoint if T == "F": return 26 if T == "G": return 7 def statistic_alternative(C): n = C.rank() T = C.type() W = WeylCharacterRing(C) for r in W.positive_roots(): if r.is_dominant() and r.is_short_root(): return W(r).degree() return W.adjoint_representation().degree()

Created

Apr 19, 2018 at 13:25 by

**Martin Rubey**Updated

Apr 19, 2018 at 14:48 by

**Martin Rubey**searching the database

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