Schur modules

Consider an n-dimensional vector space W . In this paragraph we are looking

at the m-fold tensor product W

⊗d

_{, on which the general linear group GL(W )}

and the symmetric group S

d

are acting via:

g(w

1

⊗ . . . ⊗ w

d

) := gw

1

⊗ . . . ⊗ gw

d

,

where g ∈ GL(W ),

σ(w

1

⊗ . . . ⊗ w

d

) := w

σ−1₍₁₎

⊗ . . . ⊗ w

_σ−1_(d)

,

where σ ∈ S

_d

,

extended linearly to the entire tensor product. There is a correspondence be-

tween the irreducible representations of GL(W ) and the irreducible representa-

tions of S

d

, which was discovered by Schur. We rst introduce the exterior and

symmetric powers of W .

Exterior powers ([FH04, Appendix B.2]). The exterior power V

d

W

of

the vector space W is the quotient space of W

⊗d

_{by the subspace spanned by}

all w

1

⊗ . . . ⊗ w

d

− (−1)

sgn σ

w

σ(1)

⊗ . . . ⊗ w

σ(d)

with σ ∈ S

d

. We denote the

coset of w

1

⊗ . . . ⊗ w

d

by w

1

∧ . . . ∧ w

d

. Dene V

0

W

to be the ground eld.

If {b

i

}

i=1,n

is a basis for W , then {b

i1

∧ . . . ∧ b

id

| i

1

< . . . < i

d

}

is a basis for

V

d

_W.

Symmetric powers ([FH04, Appendix B.2]). The symmetric power S

d

_W

of the vector space W is the quotient space of W

⊗d

_{by the subspace spanned}

by all w

1

⊗ . . . ⊗ w

d

− w

σ(1)

⊗ . . . ⊗ w

σ(d)

with σ ∈ S

d

. We denote the coset of

w

1

⊗ . . . ⊗ w

d

by w

1

· . . . · w

d

. Dene S

0

W

to be the ground eld. If {b

i

}

i=1,n

is

a basis for W , then {b

i1

1

· . . . · b

inn

| i

1

+ . . . + i

n

= d}is a basis for S

d

W, hence

we can see this space as the space of homogeneous polynomials of degree d in

the variables b

i

.

Schur modules ([FH04, Chap. 6]). Consider an integer d ≥ 1. We call λ =

(λ

1

, . . . , λ

k

)a partition of d if λ

1

≥ . . . λ

k

≥ 1and d = λ

1

+. . .+λ

k

. The number

of irreducible representations of S

d

coincides with the number of partitions of d

([FH04, Chap. 4]). One can associate to each partition λ = (λ

1

, . . . , λ

k

)of d a

diagram of type

λ

1

λ

2

λ

3

with λ

i

boxes in the i-th row. For example, the diagram above corresponds to

the partition (4, 3, 2, 1) of 10. These kind of diagrams are called Young diagrams.

The number of rows in the Young diagram of the partition λ of d is called the

height of λ and denoted |λ|.

By numbering the boxes in a Young diagram by the integers 1, 2, . . . , d, we

obtain a tableau of shape the given Young diagram. For example, with the Young

diagram above we get

1 2 3 4

5 6 7

8 9

10

Given a partition λ of d and a tableau of the Young diagram associated to λ,

one can dene the following two subgroups of S

d

:

P

λ

= {σ ∈ S

d

| σ

preserves each row}

Q

λ

= {σ ∈ S

d

| σ

preserves each column}

Each σ ∈ S

d

is associated to basis element e

σ

in the group algebra of S

d

(the

structure of algebra is given by e

σ1

· e

σ2

= e

σ1σ2

). Dene

c

λ

= (

X

σ∈Pλ

e

σ

) · (

X

σ∈Qλ

sgn(σ)e

σ

),

and S

λ

W = Im(c

λ

|

W⊗d

).

The subspaces S

λ

W

of W

⊗d

obtained in this way are called Schur modules.

Example 2.8.1. If λ = (2, 1) is a partition of 3, the tableau associated to it is

1 2

3

We have c

λ

= (e

1

+ e

(12)

) · (e

1

− e

(13)

) = e

1

+ e

(12)

− e

(13)

− e

(312)

and S

λ

W

is

the subspace of W

⊗3

_{spanned by all}

w

1

⊗ w

2

⊗ w

3

+ w

2

⊗ w

1

⊗ w

3

− w

3

⊗ w

2

⊗ w

1

− w

3

⊗ w

1

⊗ w

2

.

If λ = (d), then S

λ

W

is the subspace of W

⊗d

spanned by P

_σ∈Sd

w

σ(1)

⊗ . . . ⊗

w

σ(d)

, hence S

λ

W ∼= S

d

W.

If λ = (1, . . . , 1) with |λ| = d, then S

λ

W

is the subspace of W

⊗d

spanned by all

P

σ∈Sd

sgn(σ)w

σ(1)

⊗ . . . ⊗ w

σ(d)

, hence S

λ

W ∼=

V

d

W.

Theorem 2.8.1. ([Pro07, Chap. 8.1]) Consider a nite-dimensional complex

vector space W with dim W = n. Then:

(1) The list of irreducible representations of GL(W ) is

S

λ

W ⊗ (

^

n

(2) The list of irreducible representations of SL(W ) is

S

λ

W,

|λ| ≤ n − 1.

Theorem 2.8.2. ([FH04, Chap. 6]) Consider a nite-dimensional complex

vector space W with dim W = n. Then:

(1) S

λ

W

is zero if the partition λ is of the form (λ

1

, . . . , λ

d

), with d > n.

Otherwise

dim S

λ

W =

Y

1≤i<j≤n

λ

i

− λ

j

+ j − i

j − i

.

(2.3)

In particular, dim S

d

_{W =}

n+d−1 d

_{and dim V}

d

W =

n_d

_.

(2) If m

λ

is the dimension of the irreducible representation of S

d

corresponding

to λ, then

W

⊗d

'M

λ

m

λ

S

λ

W.

Example 2.8.2. We have the following decompositions:

W ⊗ W = S

2

W ⊕^

2

W,

W ⊗ W ⊗ W = S

3

W ⊕ 2S

(2,1)

W ⊕

^

3

W,

(see [FH04, Chap. 6]). With the formula2.3, dim S

3

_{W =}

n+2

3

, dim S

(2,1)

W =

2

n+1₃

_{, and dim V}

3

W =

n₃

_.

Theorem 2.8.3. ([FH04, Chap. 6]) Consider two nite-dimensional complex

vector spaces W

1

and W

2

. Then,

S

d

(W

1

⊕ W

2

) =

M

a+b=d

(S

a

W

1

⊗ S

b

W

2

);

^

d

(W

1

⊕ W

2

) =

M

a+b=d

(^

a

W

1

⊗

^

b

W

2

);

S

d

(W

1

⊗ W

2

) =

M

|λ|=d

S

λ

W

1

⊗ S

λ

W

2

;

^

d

(W

1

⊗ W

2

) =

M

|λ|=d

S

λ

W

1

⊗ S

λ0

W

₂

,

where λ

0

_{is the conjugate partition of λ, obtained by interchanging rows and}

columns in the Young diagram corresponding to λ.

Example 2.8.3. If W

1

and W

2

are two nite-dimensional vector spaces,

S

2

(W

1

⊕ W

2

) = S

2

W

2

⊕ (W

1

⊗ W

2

) ⊕ S

2

W

1

;

^

2

(W

1

⊕ W

2

) =

^

2

W

2

⊕ (W

1

⊗ W

2

) ⊕

^

2

W

1

;

S

2

(W

1

⊗ W

2

) = (S

2

W

1

⊗ S

2

W

2

) ⊕ (

^

2

W

1

⊗

^

2

W

2

);

^

2

(W

1

⊗ W

2

) = (S

2

W

1

⊗

^

2

W

2

) ⊕ (

^

2

W

1

⊗ S

2

W

2

).

Remark 2.8.4. Consider the vector space V

n

of binary forms of degree n. Then

we identify V

n

with S

n

W

∗

, where W = C

2

. The modules S

d

V

n

= S

d

(S

n

W

∗

)

and V

d

V

n

=V

d

(S

n

_W

∗

_{)decompose into irreducible representations of SL}

2

. For

example,

S

4

V

6

= 2V

0

+ 2V

4

+ V

6

+ 3V

8

+ V

10

+ 3V

12

+ V

14

+ 2V

16

+ V

18

+ V

20

+ V

24

,

^

4

V

6

= V

0

+ V

4

+ V

6

+ V

8

+ V

12

.

(We computed these decompositions using the program LiE [LCL92].)

Proposition 2.8.5. [Pro07, Chap. 15] The space of covariants of V

n

of degree

dand order e is the sum of all irreducible representations of S

d

V

_n∗

of type V

e

.

In particular, the dimension of the vector space of covariants of degree d and

order e of V

n

equals the multiplicity with which V

e

appears in the decomposition

of S

d

_V

∗ n

.

Example 2.8.4. Consider f ∈ V

3

. We have:

S

1

(V

3

) = V

3

,

S

2

(V

3

) = V

2

+ V

6

,

S

3

(V

3

) = V

3

+ V

5

+ V

9

,

S

4

(V

3

) = V

0

+ V

4

+ V

6

+ V

8

+ V

12

.

(We computed these decompositions using the program LiE [LCL92].)

In other words, the vector space of covariants of V

3

of degree 1 and order 3

has dimension 1 (spanned by f itself), the vector space of covariants of V

3

of

degree 2 and order 2 has dimension 1 (spanned by (f, f)

2

), the vector space of

invariants of V

3

of degree 2 has dimension 1 (spanned by ((f, f)

2

, (f, f )

2

)

2

), etc.

In document Invariants of binary forms (Page 34-37)