UNDER CONSTRUCTION (June 4, 2022)
Symmetric group representation theory
In hand written notes, I describe the Frobenius characteristic map
and the correspondence to symmetric functions. Specifically, the
irreducible character X_\lambda of S_n goes to s_{\lambda}, the Schur
function. See those notes.
Judy Chiang also told you about the Murnaghan-Nakayama rule (see also
the rule of Fomin-Greene) for computing character values of irreducibles.
This rule is completely combinatorial and you can look it up here:
https://en.wikipedia.org/wiki/Murnaghanâ€“Nakayama_rule
The proof can be found in Stanley's Enumerative Combinatorics 2, chapter 7.
Once you accept that X_{\lambda} maps to s_{\lambda}, the character value
of X_{\lambda}(\mu) is
z_{\mu} x the coefficient of p_{\mu} is s_{\lambda}.
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You also know that S_n irreps are indexed by partitions of size n
(this follows from general finite group rep theory + the fact that the
conjugacy classes of S_n are indexed by partitions).
You've *heard* that GL_n irreps are indexed by partitions of length <=n.
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The goal of the notes below is to describe the Young symmetrizers. This
will give a way to construct the irreps of S_n AND (essentially all) irreps
of GL_d!
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Fix a superstandard Young tableau, e.g., T_{\lambda}=
1234
56
78
9
of shape \lambda=(4,2,2,1). (That is, you put 1,2,3,4, in the obvious
reading order.)
We now define two subgroups of the symmetric group S_n (n=9 here).
Row symmetrizers
----------------
P=P_{\lambda} = {g in S_n: g preserves each row}
Q=Q_{\lambda} = {g in S_n: g preserves each column}
Ex: n=3, \lambda=(2,1)
T_{(2,1)} =
12
3
P_{21} = {(12)(3), id}
Q_{21} = {(13)(2), id}
RECALL the group algebra C[S_n] has vector space basis given by
{e_g : g in S_n} with multiplication e_g*e_{g'} = e_{gg'}.
Define elements of C[S_n] by
a_{\lambda} = \sum_{g in P_{\lambda}} e_g
b_{\lambda} = \sum_{g in Q_{\lambda}} sgn(g) e_g
Ex. In our case
a_{21} = e_1 + e_{(12)}
b_{21} = e_1 + e_{(13)}
Definition: The *Young symmetrizer* for \lambda is
c_{\lambda} = a_{\lambda}*b_{\lambda} in C[S_n].
Let
S^{\lambda}:=C[S_n]c_{\lambda}.
This is by definition an C[S_n] module (by left multiplication).
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THEOREM: S^{\lambda} is an irreducible representation of S_n; all
are distinct, and hence these are all of them.
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(The proof isn't that bad; I'll delay it for now but it can be found
in Fulton-Harris Lecture 4.)
Ex. For \lambda=(2,1)
c_{21} = (e_1+e_{(12)})(e_1 - e_{13})
= e_1 +e_{(12)}-e_{(13)} - e_{(132)}
Look at
e_1 * c_{21} = c_{21}
e_{(12)} * c_{21} = e_{(12)} + e_1 - e_{132} -e_{13} =c_{21}
e_{(13)} * c_{21} = e_{(13)} +e_{(123)} - e_1 - e_{(23)} <- NEW! "d"
e_{(123)} * c_{21} = e_{123} + e_{(13)} - e_{(23)} - e_id = d
e_{(132)} * c_{21} = e_{(132)} + e_{(23)} -e_{(12)} - e_{(123)} = -(d+c_{21})
Hence S^{(21)} is spanned by c_{21} and d.
Moreover, it is irreducible. Indeed, d=e_{(13)}*c_{21} so the S_n orbit
of c_{21} spans the entire space.
This is a degree 2 irreducible representation of S_3. By our character
table for S_3, this MUST be (isomorphic to) the standard representation.
Remarks: S^{\lambda} is called the Specht module. It has a number of
constructions:
* One of them is in Sagan, chapter 2 which is the "tabloid" approach.
It has the advantage of allowing one to talk about a basis indexed
by SYT => the degree of the representation is f^{\lambda}=#SYT of shape
\lambda.
* Another construction (the "Vandermonde construction") is to take the
subspace of C[x1,...,xn] spanned by all products
P_T:=\prod_{iEnd(V^{\otimes n})
be the correspondong map.
Special case: \lambda=(2)
**Look at permute(a_(2)). Any element x@y of V^{\otimes 2} is sent to
x@y+y@x. In other words, permute(a_(2)) PROJECTS V^{\otimes 2} to the
subspace Sym^2(V) spanned by vectors of the form x@y+y@x.
(As I will have explained, this subspace description of Sym^(V) is
isomorphic as vector spaces to the description in terms of quotienting
the tensor algebra, over characteristic 0.)
**Look at permute(b_(1,1)). Any element x@y of V^{\otimes 2} is sent
to x@y - y@x. That is permute(b_(2)) PROJECTS V^{\otimes 2} to
the subspace Alt^2(V) spanned by vectors of the form x@y - y@x.
**Notice that our simple choice of partitions (2), (1,1) implies
permute(c_(2))=permute(a_(2)) (since c_2= a_2)
permute(b_(1,1))=permute(a_(1,1)) (since b_{1,1}=a_{1,1})
**Notice more generally permute(c_(n)) projects V^{\otimes n} to the
irrep subspace Sym^n(V)
**Notice more generally permute(c_(1,1,1,1,...1,))
projects V^{\otimes n} to the irrep subspace Alt^n(V) (=Wedge^n V).
The following result is due to Weyl but maybe also Schur. Anyway, it is
called Weyl's construction or the Schur functor construction.
Theorem: permute(c_{\lambda}) projects V^{\otimes n} to V_{\lambda}, the
irreducible (polynomial) representation of GL_n indexed by a partition
with \lambda with n boxes.
The theorem is *slightly* worse than what you want. In fact it is true
that there is an irreducible for every partition **with at most n
rows**. But the idea is that once you have Weyl's construction, you can
obtain all irreps of GL_d by varying n. This is the meaning of saying
Weyl's theorem gives "essentially all irreps of GL_d".
Schur-Weyl duality: the relationship between S_n and GL_d irreps.
V^{\otimes n} has a left S_n action and a right GL_d action. Moreover,
these actions commute. This means V^{\otimes n} is a S_n x GL_d module.
Now, let's take for granted that "S_n rep theory is easier" (if only because
we know finite group theory tells us useful things, such as the number of
irreps). On the other hand, we know the irreducibles of S_n x GL_d are
of the form S^{\lambda} @ V^{\mu}. What Schur-Weyl duality tells you is
that
V^{\otimes n} iso (as S_n x GL_d modules) to
\bigoplus_{\lambda partition of n} S^{\lambda}@V^{\lambda}.
[This is the BIG GENERALIZATION of the symmetric square decomposition
V^{\otimes 2} = Sym^2(V) \oplus Alt^2(V)]
Here comes the key Schur lemma arugment. OK, I want to pick off
V^{\lambda}. First think about S_n-morphisms of S^{\lambda} to V^{\otimes n}
(treating the "@V^{\lambda}" like a multiplicity). This is about
Hom_{S_n}(S^{\lambda},V^{\otimes n})
By Schur's lemma,
V^{\lambda}=Hom_{S_n}(S^{\lambda},V^{\otimes n})
iso (S^{\lambda})^*\otimes_{C[S_n]} V^{\otimes n}
iso (S^{\lambda}) \otimes_{C[S_n]} V^{\otimes n}
Here we used that S^{\lambda} is isomorphic to S^{\lambda}^* (proof:
look at the characters, X(g)^* = complex conj(X(g)) = X(g) in the the
case of S_n.)
= C[S_n]c_{\lambda} \otimes_{C[S_n]} V^{\otimes n}
iso V^{\otimes n} c_{\lambda} [exer]
Unfortunately, the description of GL_n irreps from this viewpoint is
much worse than that for S_n (forgetting even the proofs, you can see
above at least simple enough to absorb DESCRIPTIONS of the Specht
module). See Section 15.5 of Fulton-Harris "More remarks on Weyl's
construction".
"Much worse"? For Specht modules, we say it's the span some some bunch of
vectors.
For Weyl modules, you look at a subspace of a subspace
Sym^{a_k}(Wedge^k V) @ Sym^{a_{k-1}}(Wedge^{k-1} V) @
....@ Sym^{a_1}(V)
of V^{\otimes n},
where a_i is the number of columns of lambda of length i. (NOW one
has to interpret how is is a subspace of V^{\otimes n} to start off
with; that's not too bad.) But now you need to quotient by some relations...
Remarks: See Exer 15.57 of Fulton-Harris: On the other hand, there's a
simple description of GL_n irreps done another way:
Let X=(x_{ij}) be the n x n matrix of indeterminates (i.e., C[X] is the
coordinate ring of gl_n). GL_n acts on X by linear substitutions, i.e.,
g.x_{ij} = \sum_{k=1}^ a_{k,i} x_{k,j} where g in GL_n(C).
For any tableau T of shape \lambda that is strictly increasing along columns,
let e_T be the product of minors constructed from X by:
taking for each column i of T of length \mu_i,
the rows of e_T to be 1,2,...,\mu_i and columns are the LABELS
of T in the column.
Now take D_{\lambda} to be
the span of these e_T's. The e_T's in fact form a basis and
D_{\lambda} is isomorphic to V_{\lambda}.
TO DO: Proof of the theorem on Specht modules
TO DO: More on GL_n (in a future lecture) based in part on
https://math.mit.edu/~rstan/pubs/pubfiles/57.pdf
References
Fulton and Harris' book, Lecture 4
Sagan Chapter 2
http://www.math.columbia.edu/~ums/Finite%20Group%20Rep%20Theory7.pdf