Unipotent Representation Explorer

Unipotent Representations of Classical Groups Explorer

Explore DRC diagrams, local systems, counting formulas, Barbasch–Vogan duality, and the Springer correspondence for classical groups.

Partition (comma-separated, non-increasing)

Root system typeC — D — M — B —

Type: $\mathrm{Sp}(2n)$, symplectic

DRC Diagrams

Partition type:

Found 29 DRC diagrams

#1

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(6, 7)$

rd

c

d

×

s

s

#2

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(6, 7)$

•d

c

d

×

•

s

#3

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(5, 8)$

rd

r

d

×

s

s

#4

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(5, 8)$

•d

r

d

×

•

s

#5

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(5, 8)$

•d

•

d

×

•

•

#6

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(8, 5)$

rc

c

d

×

s

s

#7

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(8, 5)$

•c

c

d

×

•

s

#8

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(7, 6)$

•r

c

d

×

•

s

#9

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(7, 6)$

rc

r

d

×

s

s

#10

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(7, 6)$

•c

r

d

×

•

s

#11

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(7, 6)$

•c

•

d

×

•

•

#12

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(6, 7)$

•r

r

d

×

•

s

#13

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(6, 7)$

•r

•

d

×

•

•

#14

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(7, 6)$

rd

r

c

×

s

s

#15

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(7, 6)$

•d

r

c

×

•

s

#16

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(7, 6)$

•d

•

c

×

•

•

#17

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(6, 7)$

rd

r

r

×

s

s

#18

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(6, 7)$

•d

r

r

×

•

s

#19

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(6, 7)$

•d

•

r

×

•

•

#20

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(9, 4)$

rc

r

c

×

s

s

#21

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(9, 4)$

•c

r

c

×

•

s

#22

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(9, 4)$

•c

•

c

×

•

•

#23

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(8, 5)$

•r

r

c

×

•

s

#24

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(8, 5)$

•r

•

c

×

•

•

#25

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(8, 5)$

rc

r

r

×

s

s

#26

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(8, 5)$

•c

r

r

×

•

s

#27

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(8, 5)$

•c

•

r

×

•

•

#28

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(7, 6)$

•r

r

r

×

•

s

#29

$G$: $\mathrm{Sp}(12)$ | $\check{G}$: $\mathrm{O}(7, 6)$

•r

•

r

×

•

•

Local Systems

Found 56 local systems

+positive (p-type)-negative (n-type)*positive (q-type)=negative (q-type)

#1

+-+-

+-+-

*=

*=

*=

signature: (7, 7)

#2

*=*=

*=*=

*=

*=

*=

signature: (7, 7)

#3

*=*=

*=*=

+-

+-

+-

|

*=*=

*=*=

-+

+-

+-

signature: (7, 7)

#4

-+-+

+-+-

+-

+-

+-

signature: (7, 7)

#5

+-+-

+-+-

+-

+-

+-

|

+-+-

+-+-

-+

+-

+-

signature: (7, 7)

#6

-+-+

*=*=

+-

+-

+-

signature: (7, 7)

#7

=*=*

+-+-

+-

+-

+-

signature: (7, 7)

#8

=*=*

*=*=

+-

+-

+-

signature: (7, 7)

#9

+-+-

+-+-

=*

*=

*=

|

+-+-

+-+-

=*

=*

*=

signature: (7, 7)

#10

=*=*

*=*=

=*

*=

*=

|

=*=*

*=*=

*=

*=

*=

signature: (7, 7)

#11

-+-+

-+-+

*=

*=

*=

signature: (7, 7)

#12

*=*=

*=*=

=*

*=

*=

|

*=*=

*=*=

=*

=*

*=

signature: (7, 7)

#13

=*=*

+-+-

=*

*=

*=

|

=*=*

+-+-

*=

*=

*=

signature: (7, 7)

#14

-+-+

*=*=

=*

*=

*=

|

-+-+

*=*=

*=

*=

*=

signature: (7, 7)

#15

=*=*

=*=*

*=

*=

*=

signature: (7, 7)

#16

-+-+

+-+-

=*

*=

*=

|

-+-+

+-+-

*=

*=

*=

signature: (7, 7)

#17

*=*=

*=*=

-+

-+

+-

|

*=*=

*=*=

-+

-+

-+

signature: (7, 7)

#18

-+-+

+-+-

-+

+-

+-

|

-+-+

+-+-

-+

-+

+-

signature: (7, 7)

#19

=*=*

=*=*

+-

+-

+-

|

=*=*

=*=*

-+

+-

+-

signature: (7, 7)

#20

+-+-

+-+-

-+

-+

+-

|

+-+-

+-+-

-+

-+

-+

signature: (7, 7)

#21

-+-+

*=*=

-+

+-

+-

|

-+-+

*=*=

-+

-+

+-

signature: (7, 7)

#22

=*=*

+-+-

-+

+-

+-

|

=*=*

+-+-

-+

-+

+-

signature: (7, 7)

#23

-+-+

-+-+

+-

+-

+-

|

-+-+

-+-+

-+

+-

+-

signature: (7, 7)

#24

=*=*

*=*=

-+

+-

+-

|

=*=*

*=*=

-+

-+

+-

signature: (7, 7)

#25

+-+-

+-+-

=*

=*

=*

signature: (7, 7)

#26

=*=*

*=*=

=*

=*

*=

|

=*=*

*=*=

=*

=*

=*

signature: (7, 7)

#27

-+-+

-+-+

=*

*=

*=

|

-+-+

-+-+

=*

=*

*=

signature: (7, 7)

#28

*=*=

*=*=

=*

=*

=*

signature: (7, 7)

#29

=*=*

+-+-

=*

=*

*=

|

=*=*

+-+-

=*

=*

=*

signature: (7, 7)

#30

-+-+

*=*=

=*

=*

*=

|

-+-+

*=*=

=*

=*

=*

signature: (7, 7)

#31

=*=*

=*=*

=*

*=

*=

|

=*=*

=*=*

=*

=*

*=

signature: (7, 7)

#32

-+-+

+-+-

=*

=*

*=

|

-+-+

+-+-

=*

=*

=*

signature: (7, 7)

#33

-+-+

+-+-

-+

-+

-+

signature: (7, 7)

#34

=*=*

=*=*

-+

-+

+-

|

=*=*

=*=*

-+

-+

-+

signature: (7, 7)

#35

-+-+

*=*=

-+

-+

-+

signature: (7, 7)

#36

=*=*

+-+-

-+

-+

-+

signature: (7, 7)

#37

-+-+

-+-+

-+

-+

+-

|

-+-+

-+-+

-+

-+

-+

signature: (7, 7)

#38

=*=*

*=*=

-+

-+

-+

signature: (7, 7)

#39

-+-+

-+-+

=*

=*

=*

signature: (7, 7)

#40

=*=*

=*=*

=*

=*

=*

signature: (7, 7)

#41

+-+-

+-+-

=*

+-

+-

signature: (7, 7)

#42

*=*=

*=*=

=*

+-

+-

signature: (7, 7)

#43

*=*=

*=*=

-+

-+

*=

signature: (7, 7)

#44

=*=*

+-+-

-+

*=

*=

signature: (7, 7)

#45

+-+-

+-+-

-+

-+

*=

signature: (7, 7)

#46

=*=*

*=*=

-+

*=

*=

signature: (7, 7)

#47

-+-+

+-+-

-+

*=

*=

signature: (7, 7)

#48

-+-+

*=*=

-+

*=

*=

signature: (7, 7)

#49

-+-+

*=*=

=*

=*

+-

signature: (7, 7)

#50

=*=*

=*=*

=*

+-

+-

signature: (7, 7)

#51

-+-+

+-+-

=*

=*

+-

signature: (7, 7)

#52

=*=*

*=*=

=*

=*

+-

signature: (7, 7)

#53

-+-+

-+-+

=*

+-

+-

signature: (7, 7)

#54

=*=*

+-+-

=*

=*

+-

signature: (7, 7)

#55

-+-+

-+-+

-+

-+

*=

signature: (7, 7)

#56

=*=*

=*=*

-+

-+

*=

signature: (7, 7)

Count Unipotent Representations

Root type:

Selected type: $\mathrm{Sp}(2n)$ — types $B$, $D$, $CS$ return a polynomial in $p$, $q$; types $C$, $M$, $DS$ return an integer.

0

Barbasch–Vogan Duality

partrc:

BV dual partition

(4, 4, 2, 2, 1, 1)size: 14

Springer Correspondence

W-representation $(\tau_L,\, \tau_R)$

$\tau_L = (1, 2)$|$\tau_R = (1, 3)$

Round-trip

(5, 5, 2, 2)

Theta Lifting Graph

Mode:

Loading Graphviz renderer…

Descent lift (l)Generalized descent lift (L)Twist (1,−1) [p]Twist (−1,1) [q]Det twist (−1,−1) [d]