Post on 05-Aug-2020
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.