# Nilpotent conjugacy classes in the classical groups

(1)

n

n

◦ ◦

◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

4

n

ν

(2)

i

1

2

k

i

k

1

2

k

1

2

2

3

DOMINANCE

k

λ

λ k

λ k

k 1

i

λ

k

n

k=0

0

n

k

(3)

λ

k

1

k

µ

k

1

k

λ

µ

(4)

SHIFTS

i,j

k

i

j

i,j

i,j

i,j

2,4

µ

k

λ

k

λ

k

λ

k

i,j

i+1

i

j

j−1

i+1

i

k

λ k

µ

k

i,j

i,j

i+1

i

j

i

k

i

(5)

k,ℓ

λ k

µ

k

k+1

k

i,i+1

λ m

µ

m

λ

i

µ

i

λ

m

µ

m

i

i

m

m

m

µ

m

λ

m

m

m

m

k

m

k+1

k

λ

k

µ

k

λ

k

µ

k

k

m

m

k

λ

k

λ

m

m

λ

m

m

m

m

µ

m

m

µ

m

k+1

m

µ

k

k+1

k

m,m+1

(6)

k+1

k

k+1

ℓ+1

i

k

λ ℓ

µ

ℓ+1

ℓ,ℓ+1

ℓ+1

k

k,ℓ+1

i

j

## λ µ κ

### −→ −→

THE LATTICE OF PARTITIONS

λ

µ

## λ ∧ µ

(7)

i

i

1

2

i

j

i

i+1

j

i

i

i−1

1

λi

λi

1

j

j−1

j

(8)

### In this section I’ll begin to explain the relationship between partitions and nilpotent matrices.

STANDARD NILPOTENT MATRICES

i,j

i,j

4

◦ ◦

◦ ◦ ◦

◦ ◦ ◦ ◦

λ

λi

3,2

λ

EXISTENCE

n

n

i

n

n

n

i

i

i

i

n

N

n

n

N

n

n

λi

1

k

1

k

ℓ−1

λ

k

k−1

k−1

i

i

i

i

k

k

(9)

UNIQUENESS

n

0

1

n

n

k

k

k−1

1

2

λ

1 2

3 4

5 6

λ

6

3

5

1

4

2

ti,j

ti,j−1

k

λ

i

j

j

i

(10)

λ

λ

λ

n

i

i

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦ ◦ ◦

◦ ◦ ◦ ◦ ◦ ◦

◦ ◦ ◦ ◦ ◦ ◦ ◦

## 

λ

λ

n

n

λ

λ

λ

λ

kλ

kλ

k

j

1

k

λ

λ

n

λ

λ

µ

λ

1

n−1

x

i,j

i

x

−1

y

i

i

i

i

i,i

i+1,i+1

λ

λ

(11)

λ

λ

λ

k

k

1

k

m

n−k+1

n−¯λk+1

λ

µ

λ

µ

λ

λ

κ

µ

κ

µ

λ

λ

µ

k

k−1

k

k−1

1 2 3 4

5 6 7

8 9 10

11 12

13 14

15 16

17 1

2 3 4

5 6 7

8 9 10

11 12

13 14

15

16

17

i

m

n

i

n

c−1

k

k−1

j

k

k−j

k

k

k

k

k

k

(12)

ν

n

ν

n

T

λ

λ

T

λ

### The proof will require a serious digression. In compensation for its length, other interesting matters will come to light.

LISTING YOUNG TABLEAUX

FILTRATIONS OF N

d

d−1

1

0

i

i−1

i

i−1

i

i

a

i−1

a

i−1

i

i

i

i

1

1

2

2

d

d

i

j

k>i,j

k

(13)

i

i

2

2

1

1

1

1

2

2

2

1

d−1

d

d

d

d

n

n

ROOTS

n

i,j

i,j

−1

i

j

i,j

n

i,j

i

j

i,j

i

j

i

i

n

λ

λ

λ

λ

λ

i,j

i,j

k,ℓ

i,ℓ

λ

µ

λ+µ

λ+µ

λ

µ

−1λ

µ

λ

µ

λ

µ

µ

λ

λ+µ

µ

λ

λ+µ

µ

λ

(14)

≥λ

µ≥λ

µ

n

7 8 9 10

4 5 6

2 3 1

## 1

i,j

n−1,n

n−1,n

n−2,n−1

n−2,n−1

n−2,n

n−2,n

1,n

1,n

2

3

1

1

2

3

(15)

2

FLAGS

0

1

n−1

n

k

k−1

n

n

n

k

i

n

i

i

i

i

i

k

k−1

i

U

U

i

j

i

i

i

i

i+1

i

1

2

n

k

1

k

(16)

k

3

2

4

1

4

1

4

2

4

3

4

4

4

5

i

i

σ(i)

n−1

n

−1

−1

w

−1

w

−1

λ>0,wλ<0

λ

w

λ>0,wλ>0

λ

w

w

w

−1

w

w

w

w

xy

y

−1

x

x

y

xy

w

w

−1w

−1

w

w

0 xy

2

1

0 x

1

1 x

2

SPRINGER FIBERS

g

i

i si

i+1

0

n

u

i

s

u

u

u

u

k

(17)

3,2,1

1

2

3

4

5

6

1 2

3

4 5 6

ν

ν

1

2

3

2

1

2

1

3

1

2

1

2

3

1

1

ν

1 3 2

1 2 3

1

1

k

u

u

i

i

u

(18)

k−1

k

k

k−1

k

k−1

k

j

u

k

k

CENTRALIZERS

ν

ν

G

n

5,3,3

5

3

3

n

1,1

1,2

1,3

2,1

2,2

2,3

3,1

3,2

3,3

5

3

3

5

3

3

1,1

1,2

1,3

2,1

2,2

2,3

3,1

3,2

3,3

i,j

i

j

i,j

1

5

2

3

3

3

i,j

5

3

5

3

λ

i,j

i

j

λ

n

2i

(19)

2

2

2

2

2

G

n

G

### (ν) is connected.

CONCLUDING THE PROOF OF THE THEOREM

u

−1

G

u

u

G

u

u

(20)

r

r

−1

−1

t

m

t

m

m

m

m

m

m

(21)

n

◦ ◦

◦ ◦

n

n

n

n

n

n

n

2n

2n

2n+1

t

−1

t

−1

t

m

n

n

t

t

t

t

n

n

n

n

2n

(22)

2n

t

−1

n

2n

n

2n

t

−1

−1

2n

t

−1

i

2n

i

2n

2n

2n

SUFFICIENCY.

n

2n

n

n

k

k

2k

2ℓ

2n

k

k

k

k

k

k

(2n)

2n

n

n,n

n

2n

n

p,q

4

◦ ◦

◦ ◦ ◦

◦ ◦ ◦ ◦

(23)

2

◦ ◦

◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

3

◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

4

◦ ◦

◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

◦ ◦

◦ ◦ ◦ ◦

◦ ◦ ◦ ◦

NECESSITY.

2n

2n

2n

n

n

2n+1

2n+1

n

n

2n

t

(24)

2n+1

t

t

t

k

k+ℓ

2k+1

2(k+1)

2

2

t

1

2

t

2

2

t

t

1

t

1

−1

2

1

2n+1

2n+2

i

m

i

m

m

m

m

m

SUFFICIENCY.

n

k

k+ℓ

(2n+1)

2n+1

2n+1,2n

2n

2n+1

5

2n

2n

2n

2n

2n

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