BTW:
naturally have
sl V -> gl V -> K
this splits by "divide by n"
(that is, the scalar matrices splitting yields *n)
SL V -> GL V -> K^*
this wants to split by "nth root"
(that is, the scalar matrices splitting yields ^n)
...but there's in general an obstruction!
(algebraically, there's not a unique nth root;
topologically, )
over R, you can do this in odd dimension,
or for GL^+ in even dimension
(so you get unique roots)
gl V = K \oplus sl V
(for p \ndivides n)
concretely,
the internal projection is \frac{1}{n} \epsilon\eta x
so
x \mapsto (\frac{1}{n} \epsilon\eta x, x - \frac{1}{n} \epsilon\eta x)
confusingly:
GL V < gl V
Aut V < End V
from the Lie POV, this is *weird*
...but categorically natural
gl V -> GL V via exp
End V -> Aut V
from the categorical POV, this is *weird*
...but Lie natural
[btw, you do get det on the whole endomorphism semigroup
SL V -> End V -> K
...which is also a bit weird]
oh, BTW, these maps are onto; use a rank 1 matrix, concretely
(1 0)
(0 0)
e^i \otimes e_i
...if you wanna be abstract about it
btw, \Lambda^0 V = {0}
while \Lambda^{-1} = \emptyset
(yeah, I love it)
trace is more fundamental (comes from identity and adjunction);
det is special (comes from $\Lambda^n$)
also have a section on symmetric polys and Schur functors
. symmetric polys:
elementary sym polys are just coefficients of \prod (1+x_i)
(don't forget \sigma_0 = 1 !)
(and if you want, \sigma_{-n} = 0, sorta)
hmmm...if you want, can use \exp (1+x_i) to get all orders...
(as in Chern character)
they are functions on K^n/Sigma^n
...but also End V/Aut V (conjugation action)
[you recover rank from 'em, for instance]
trace and det: exterior powers and symmetric polynomials
coefficients of char poly
--------------------------------------------------
trace and det
gl and GL; Aut and End
(gl -> GL via exp; Aut < End; GL < gl is weird writing)
lie algebra
(gl = sl + k (scalars))
SL -> GL -> K^*
[...and Lie group pov; there can't split b/c of topology]
\section{sl}
traceless = sl, the most basic lie algebra
btw, write as sl_{k+1};
sl_1 = 0 !!
practical uses of Schur functors:
get representations in representation theory!
coefficients of char poly
Multi-linear algebra: Schur functors and Schur-Weyl duality
\chapter{Trace and Determinant: a Lie theory perspective}
Geometrically, the trace is the infinitesimal change in volume: $\tr = \det'_I$.
Trace is the derivative of the determinant at the identity.
The determinant is a map of Lie groups, and the trace is a map of Lie algebras,
so it's not surprising that they should be related in this way.
It's very easy to compute the derivative of $\det$ at the identity,
by differentiating in terms of a basis:
$$\det(I + \epsilon e^i_j) =
\begin{cases}
1 & i \neq j\\
1 + \epsilon & i = j
\end{cases}$$
It is this easy because these infinitesimals generate 1-parameter subgroups:
$\psmallmatrix{1&a\\0&1}$ is an embedding of the additive group $K^+$,
while $\psmallmatrix{a&0\\0&1}$ (for $a\neq 0$) is an embedding of the multiplicative group $K^*$.
Indeed, $\det \equiv 1$ on $U$, the group of unipotent matrices, and is just multiplication on the diagonal matrices.
which I discuss elsewhere.
related to sl + K and SL x K^*