Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Define the tensor product of U and V over F as the linear span ofsimple tensors ub v (where u P U , v P V )

U b V “ F tub v | u P U, v P V u

modulo “bilinear” relations:

pαuq b v “ αpub vq “ ub pαvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Define the tensor product of U and V over F as the linear span ofsimple tensors ub v (where u P U , v P V )

U b V “ F tub v | u P U, v P V u

modulo “bilinear” relations:

pαuq b v “ αpub vq “ ub pαvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Define the tensor product of U and V over F as the linear span ofsimple tensors ub v (where u P U , v P V )

U b V “ F tub v | u P U, v P V u

modulo “bilinear” relations:

pαuq b v “ αpub vq “ ub pαvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.

A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Define the tensor product of U and V over F as the linear span ofsimple tensors ub v (where u P U , v P V )

U b V “ F tub v | u P U, v P V u

modulo “bilinear” relations:

pαuq b v “ αpub vq “ ub pαvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsTake 1: Let F be a field, and let U and V be vector spaces over F .

Example: Let F “ R, U “ Rte1, e2u, V “ Rte1, e2, e3u.A priori, U b V seems HUGE:

U b V “ Rtpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q | ci P Ru{(relations)

“

$

’

&

’

%

ÿ

c“pc1,...,c5qPR5finite

αcpc1e1 ` c2e2q b pc3e1 ` c4e2 ` c5e5q

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

αc P R

,

/

.

/

-

{(relations).

But lots of these elements are actually the same! For example,

pe1 ` 2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1 ´ e4q ` p2e2q b p3e1 ´ e4q ` 5pe1 b e1q

“ e1 b p3e1q ` e1 b p´e4q ` p2e2q b p3e1q ` p2e2q b p´e4q ` 5pe1 b e1q

“ 3pe1 b e1q ´ e1 b e4 ` 6pe2 b e1q ´ 2pe2 b e4q ` 5pe1 b e1q

“ 8pe1 b e1q ` 6pe2 b e1q ´ e1 b e4 ´ 2pe2 b e4q

“ p8e1q b e1 ` p6e2q b e1 ´ e1 b e4 ´ p2e2q b e4

“ p8e1 ` 6e2q b e1 ´ pe1 ` 2e2q b e4 . . .

Section 10.4: Tensor productsDefine the tensor product of U and V over F as the linear span ofsimple tensors ub v (where u P U , v P V )

U b V “ F tub v | u P U, v P V u

modulo “bilinear” relations:

pαuq b v “ αpub vq “ ub pαvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Observation 1: Not every element of U b V is a simple tensor. Forexample, e1 b e2 ` e2 b e1.

Observation 2: Fix a basis B of U and C of V . . .Iteratively using bilinearity to expand simple tensors, every elementof U b V can be written as a linear combination of simple tensorsof the form eb f where e P B and f P C, i.e.

U b V “ F teb f | e P B, f P Cu{(relations).

Thm. U b V has basis teb f | e P B, f P Cu.Cor. dimpU b V q “ dimpUqdimpV q

Section 10.4: Tensor productsDefine the tensor product of U and V over F as the linear span ofsimple tensors ub v (where u P U , v P V )

U b V “ F tub v | u P U, v P V u

modulo “bilinear” relations:

pαuq b v “ αpub vq “ ub pαvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Observation 1: Not every element of U b V is a simple tensor. Forexample, e1 b e2 ` e2 b e1.

Observation 2: Fix a basis B of U and C of V . . .Iteratively using bilinearity to expand simple tensors, every elementof U b V can be written as a linear combination of simple tensorsof the form eb f where e P B and f P C

, i.e.

U b V “ F teb f | e P B, f P Cu{(relations).

Thm. U b V has basis teb f | e P B, f P Cu.Cor. dimpU b V q “ dimpUqdimpV q

Section 10.4: Tensor productsDefine the tensor product of U and V over F as the linear span ofsimple tensors ub v (where u P U , v P V )

U b V “ F tub v | u P U, v P V u

modulo “bilinear” relations:

pαuq b v “ αpub vq “ ub pαvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Observation 1: Not every element of U b V is a simple tensor. Forexample, e1 b e2 ` e2 b e1.

Observation 2: Fix a basis B of U and C of V . . .Iteratively using bilinearity to expand simple tensors, every elementof U b V can be written as a linear combination of simple tensorsof the form eb f where e P B and f P C, i.e.

U b V “ F teb f | e P B, f P Cu{(relations).

Thm. U b V has basis teb f | e P B, f P Cu.Cor. dimpU b V q “ dimpUqdimpV q

Section 10.4: Tensor productsDefine the tensor product of U and V over F as the linear span ofsimple tensors ub v (where u P U , v P V )

U b V “ F tub v | u P U, v P V u

modulo “bilinear” relations:

pαuq b v “ αpub vq “ ub pαvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Observation 1: Not every element of U b V is a simple tensor. Forexample, e1 b e2 ` e2 b e1.

Observation 2: Fix a basis B of U and C of V . . .Iteratively using bilinearity to expand simple tensors, every elementof U b V can be written as a linear combination of simple tensorsof the form eb f where e P B and f P C, i.e.

U b V “ F teb f | e P B, f P Cu{(relations).

Thm. U b V has basis teb f | e P B, f P Cu.

Cor. dimpU b V q “ dimpUqdimpV q

Section 10.4: Tensor productsDefine the tensor product of U and V over F as the linear span ofsimple tensors ub v (where u P U , v P V )

U b V “ F tub v | u P U, v P V u

modulo “bilinear” relations:

pαuq b v “ αpub vq “ ub pαvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Observation 1: Not every element of U b V is a simple tensor. Forexample, e1 b e2 ` e2 b e1.

Observation 2: Fix a basis B of U and C of V . . .Iteratively using bilinearity to expand simple tensors, every elementof U b V can be written as a linear combination of simple tensorsof the form eb f where e P B and f P C, i.e.

U b V “ F teb f | e P B, f P Cu{(relations).

Thm. U b V has basis teb f | e P B, f P Cu.Cor. dimpU b V q “ dimpUqdimpV q

Example: Let V “ Cn. S2 acts on V b V by

p12q ¨ pub vq “ v b u (and extend linearly).

You try: Let n “ 2, so V “ Cte1, e2u.1. Write the matrix for the action of p12q with respect to the

ordered basis

v1 “ e1 b e1, v2 “ e1 b e2, v4 “ e2 b e1, v2 “ e2 b e2.

2. Decompose V b V as a CS2-module.

Example: Let G be a finite group, and let U and V beCG-modules. Claim: U b V is a G-module under the action

g ¨ pub vq “ pg ¨ uq b pg ¨ vq (and extend linearly).

You try: Let R “ Ctx1, x2u be the reflection representation ofCS3. Consider the module M “ RbR.

1. Write the matrix for the action on M of p12q and p23q withrespect to the ordered basis

v1 “ x1 b x1, v2 “ x1 b x2, v4 “ x2 b x1, v2 “ x2 b x2.

2. Decompose RbR as a CS2-module.

Example: Let V “ Cn. S2 acts on V b V by

p12q ¨ pub vq “ v b u (and extend linearly).

You try: Let n “ 2, so V “ Cte1, e2u.1. Write the matrix for the action of p12q with respect to the

ordered basis

v1 “ e1 b e1, v2 “ e1 b e2, v4 “ e2 b e1, v2 “ e2 b e2.

2. Decompose V b V as a CS2-module.

Example: Let G be a finite group, and let U and V beCG-modules. Claim: U b V is a G-module under the action

g ¨ pub vq “ pg ¨ uq b pg ¨ vq (and extend linearly).

You try: Let R “ Ctx1, x2u be the reflection representation ofCS3. Consider the module M “ RbR.

1. Write the matrix for the action on M of p12q and p23q withrespect to the ordered basis

v1 “ x1 b x1, v2 “ x1 b x2, v4 “ x2 b x1, v2 “ x2 b x2.

2. Decompose RbR as a CS2-module.

Example: Let V “ Cn. S2 acts on V b V by

p12q ¨ pub vq “ v b u (and extend linearly).

You try: Let n “ 2, so V “ Cte1, e2u.1. Write the matrix for the action of p12q with respect to the

ordered basis

v1 “ e1 b e1, v2 “ e1 b e2, v4 “ e2 b e1, v2 “ e2 b e2.

2. Decompose V b V as a CS2-module.

Example: Let G be a finite group, and let U and V beCG-modules. Claim: U b V is a G-module under the action

g ¨ pub vq “ pg ¨ uq b pg ¨ vq (and extend linearly).

You try: Let R “ Ctx1, x2u be the reflection representation ofCS3. Consider the module M “ RbR.

1. Write the matrix for the action on M of p12q and p23q withrespect to the ordered basis

v1 “ x1 b x1, v2 “ x1 b x2, v4 “ x2 b x1, v2 “ x2 b x2.

2. Decompose RbR as a CS2-module.

Section 10.4: Tensor products

Take 2: Let R be a commutative ring, and let U and V beR-modules.

Define the tensor product of U and V over R as the integral spanof simple tensors ub v (where u P U , v P V )

U bR V “ Rtub v | u P U, v P V u

modulo bilinear-ish relations:

pruq b v “ rpub vq “ ub prvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Example: Z{mZbZ Z{nZ – Z{ gcdpm,nqZ.

Section 10.4: Tensor products

Take 2: Let R be a commutative ring, and let U and V beR-modules.

Define the tensor product of U and V over R as the integral spanof simple tensors ub v (where u P U , v P V )

U bR V “ Rtub v | u P U, v P V u

modulo bilinear-ish relations:

pruq b v “ rpub vq “ ub prvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Example: Z{mZbZ Z{nZ – Z{ gcdpm,nqZ.

Section 10.4: Tensor products

Take 2: Let R be a commutative ring, and let U and V beR-modules.

Define the tensor product of U and V over R as the integral spanof simple tensors ub v (where u P U , v P V )

U bR V “ Rtub v | u P U, v P V u

modulo bilinear-ish relations:

pruq b v “ rpub vq “ ub prvq

pu1 ` u1q b v “ u1 b v ` u2 b v,

ub pv1 ` v2q “ ub v1 ` ub v2.

Example: Z{mZbZ Z{nZ – Z{ gcdpm,nqZ.

Take 3: Let S Ě R be rings, and let M be a R-module.

Define the tensor product of S and M over R as the integral spanof simple tensors sbm (where s P S, m PM)

S bR M “ Stsbm | s P S,m PMu

“ IndSRpMq

modulo bilinear-ish relations:

s1 ¨ ps2 bmq “ ps1s2q bm, psrq bm “ sb pr ¨mq

ps1 ` s1q bm “ s1 bm` s2 bm,

sb pm1 `m2q “ sbm1 ` sbm2.

Idea: We want a canonical way to turn M into an S-module (toinduce an S-module). S bR M is basically the free S-span of M ,modulo relations coming from the known action of R.

Example: Let S be the sign module for S2. Compute CS3bCS2 S.

Observation: If H ď G, M is an FH-module, and tg1, . . . , g`u is aset of left coset representatives, then as sets,

gi bFH M “ giH bFH M.

Claim: If B is a basis for M , then as a vector space, pFGq bFH Mhas basis tgi b v | i “ 1, . . . , `; v P Bu.

Take 3: Let S Ě R be rings, and let M be a R-module.

Define the tensor product of S and M over R as the integral spanof simple tensors sbm (where s P S, m PM)

S bR M “ Stsbm | s P S,m PMu

“ IndSRpMq

modulo bilinear-ish relations:

s1 ¨ ps2 bmq “ ps1s2q bm, psrq bm “ sb pr ¨mq

ps1 ` s1q bm “ s1 bm` s2 bm,

sb pm1 `m2q “ sbm1 ` sbm2.

Idea: We want a canonical way to turn M into an S-module (toinduce an S-module). S bR M is basically the free S-span of M ,modulo relations coming from the known action of R.

Example: Let S be the sign module for S2. Compute CS3bCS2 S.

Observation: If H ď G, M is an FH-module, and tg1, . . . , g`u is aset of left coset representatives, then as sets,

gi bFH M “ giH bFH M.

Claim: If B is a basis for M , then as a vector space, pFGq bFH Mhas basis tgi b v | i “ 1, . . . , `; v P Bu.

Take 3: Let S Ě R be rings, and let M be a R-module.

Define the tensor product of S and M over R as the integral spanof simple tensors sbm (where s P S, m PM)

S bR M “ Stsbm | s P S,m PMu “ IndSRpMq

modulo bilinear-ish relations:

s1 ¨ ps2 bmq “ ps1s2q bm, psrq bm “ sb pr ¨mq

ps1 ` s1q bm “ s1 bm` s2 bm,

sb pm1 `m2q “ sbm1 ` sbm2.

Idea: We want a canonical way to turn M into an S-module (toinduce an S-module).

S bR M is basically the free S-span of M ,modulo relations coming from the known action of R.

Example: Let S be the sign module for S2. Compute CS3bCS2 S.

Observation: If H ď G, M is an FH-module, and tg1, . . . , g`u is aset of left coset representatives, then as sets,

gi bFH M “ giH bFH M.

Claim: If B is a basis for M , then as a vector space, pFGq bFH Mhas basis tgi b v | i “ 1, . . . , `; v P Bu.

Take 3: Let S Ě R be rings, and let M be a R-module.

Define the tensor product of S and M over R as the integral spanof simple tensors sbm (where s P S, m PM)

S bR M “ Stsbm | s P S,m PMu “ IndSRpMq

modulo bilinear-ish relations:

s1 ¨ ps2 bmq “ ps1s2q bm, psrq bm “ sb pr ¨mq

ps1 ` s1q bm “ s1 bm` s2 bm,

sb pm1 `m2q “ sbm1 ` sbm2.

Idea: We want a canonical way to turn M into an S-module (toinduce an S-module). S bR M is basically the free S-span of M ,modulo relations coming from the known action of R.

Example: Let S be the sign module for S2. Compute CS3bCS2 S.

Observation: If H ď G, M is an FH-module, and tg1, . . . , g`u is aset of left coset representatives, then as sets,

gi bFH M “ giH bFH M.

Claim: If B is a basis for M , then as a vector space, pFGq bFH Mhas basis tgi b v | i “ 1, . . . , `; v P Bu.

Take 3: Let S Ě R be rings, and let M be a R-module.

Define the tensor product of S and M over R as the integral spanof simple tensors sbm (where s P S, m PM)

S bR M “ Stsbm | s P S,m PMu “ IndSRpMq

modulo bilinear-ish relations:

s1 ¨ ps2 bmq “ ps1s2q bm, psrq bm “ sb pr ¨mq

ps1 ` s1q bm “ s1 bm` s2 bm,

sb pm1 `m2q “ sbm1 ` sbm2.

Idea: We want a canonical way to turn M into an S-module (toinduce an S-module). S bR M is basically the free S-span of M ,modulo relations coming from the known action of R.

Example: Let S be the sign module for S2. Compute CS3bCS2 S.

Observation: If H ď G, M is an FH-module, and tg1, . . . , g`u is aset of left coset representatives, then as sets,

gi bFH M “ giH bFH M.

Claim: If B is a basis for M , then as a vector space, pFGq bFH Mhas basis tgi b v | i “ 1, . . . , `; v P Bu.

Take 3: Let S Ě R be rings, and let M be a R-module.

Define the tensor product of S and M over R as the integral spanof simple tensors sbm (where s P S, m PM)

S bR M “ Stsbm | s P S,m PMu “ IndSRpMq

modulo bilinear-ish relations:

s1 ¨ ps2 bmq “ ps1s2q bm, psrq bm “ sb pr ¨mq

ps1 ` s1q bm “ s1 bm` s2 bm,

sb pm1 `m2q “ sbm1 ` sbm2.

Idea: We want a canonical way to turn M into an S-module (toinduce an S-module). S bR M is basically the free S-span of M ,modulo relations coming from the known action of R.

Example: Let S be the sign module for S2. Compute CS3bCS2 S.

Observation: If H ď G, M is an FH-module, and tg1, . . . , g`u is aset of left coset representatives, then as sets,

gi bFH M “ giH bFH M.

Claim: If B is a basis for M , then as a vector space, pFGq bFH Mhas basis tgi b v | i “ 1, . . . , `; v P Bu.

Take 3: Let S Ě R be rings, and let M be a R-module.

Define the tensor product of S and M over R as the integral spanof simple tensors sbm (where s P S, m PM)

S bR M “ Stsbm | s P S,m PMu “ IndSRpMq

modulo bilinear-ish relations:

s1 ¨ ps2 bmq “ ps1s2q bm, psrq bm “ sb pr ¨mq

ps1 ` s1q bm “ s1 bm` s2 bm,

sb pm1 `m2q “ sbm1 ` sbm2.

Idea: We want a canonical way to turn M into an S-module (toinduce an S-module). S bR M is basically the free S-span of M ,modulo relations coming from the known action of R.

Example: Let S be the sign module for S2. Compute CS3bCS2 S.

Observation: If H ď G, M is an FH-module, and tg1, . . . , g`u is aset of left coset representatives, then as sets,

gi bFH M “ giH bFH M.

Claim: If B is a basis for M , then as a vector space, pFGq bFH Mhas basis tgi b v | i “ 1, . . . , `; v P Bu.