For nonabelian groups 
The first candidate to look at is the center 
Theorem Z1. The center 
As a consequence of Theorem Z1, indicator functions of conjugacy classes in 
and because distinct conjugacy classes are disjoint we have
Theorem Z2. The class basis 
The problem we pose is to determine whether the class algebra
Theorem Z3. The class algebra 
The proof of Theorem Z3 is based on a higher-dimensional generalization of the multiplicative characters of 
where 
Then, by construction, 
This is where the key notion of 
The set 
Theorem Z4. We have
Thus our character orthogonality question becomes the question of when the space of intertwiners between two given unitary representations is zero-dimensional. To understand when this occurs, the key notion is that of a subrepresentation. A subrepresentation of a unitary representation 
Let us take a moment to note that 
such that the restriction of 
Theorem Z5. If 
A unitary representation
Theorem Z6. A nonzero
A consequence of this, which is often also called Schur’s Lemma, is the following. Note: Theorem Z6 does not require the Fundamental Theorem of Algebra, but Theorem Z7 does.
Theorem Z7. If 
There is an important consequence of Theorem Z7, which is the third and final result in the trinity of theorems collectively known as “Schur’s Lemma.” To state it, note first that there is a bijection between unitary representations of 
Theorem Z8. If 
Combining Theorems Z4,Z6, and Z7 we reach the following conclusion: characters of non-isomorphic irreducible unitary representations are orthogonal.
Theorem Z9. If 
Let 
The fact that the orthogonal set 
Conversely, given a linear representation
Theorem Z10. As a unitary representation of 
Theorem Z10 is a key result. To prove it, one first shows that for any unitary representation
Make sure you know how to do this. Then, one computes the character of the regular representation,
and combines this with the above multiplicity formula. With Theorem Z9 in hand, the next result is easily obtainable. For each 
where 
Theorem Z11. The orthogonal complement of 
The proof of Theorem Z11 uses Theorem Z9 together with the third part of Schur’s Lemma, Theorem Z8.
This completes Phase I of the construction of the Fourier transform on the class algebra
. However, the two are quite different as algebras, and to emphasize this we will write 
for the elementary function corresponding to 
rather than 
Then, the convolution product of two elementary functions in 
is the group product. This is not at all the same as the point wise product rule 
. Extending the above bilinearly, we get that the convolution of any two functions
where 
is the group identity. 
so that 
of 
is the unit circle in the complex plane. The set of all multiplicative characters is denoted 
and called the dual group of 
first recall that every finite abelian group is isomorphic to a product of cyclic groups, so we may without loss in generality assume that
is a cyclic group of order 
with generator 
Let 
so that explicitly 
is the principal 
th root of unity.
is an orthogonal set in 
it is a basis of 
are a Fourier basis of 
is an algebra isomorphism 
Since 
,
and 
which says that if one is highly concentrated, the other cannot be.
defined by
for the Fourier basis of 
of 
, and moreover that
and the eigenvalues 
in terms of the multiplicative characters 
of the underlying groups 
or the discrete hypercube 
is the Hilbert space 
containing 
as the prototypical commutative algebra and 
as the prototypical noncommutative algebra.
namely the diagonal subalgebra 
consisting of operators on 
is a scalar. Indeed, the isomorphism 
is the Fourier transform on 
, namely the basis of elementary operators 
a set of pairwise orthogonal projections that spans the diagonal subalgebra 
The Fourier transform 
is 
such that 
for some orthonormal basis 
. Typically, this is posed as a problem about matrices: you are given the matrix of 
and its Fourier transform 
outputs eigenvalues, 
Our general taxonomy of special elements in an abstract algebra gives you a number of conditions which imply that an operator is normal. 
is a maximal commutative subalgebra if and only if 
for some orthonormal basis 
for some orthonormal basis 
of 
, use the general fact that an algebra is commutative if and only if all its elements are normal, and using Theorem E1 show that there exists an orthonormal basis 
Since all subalgebras of function algebras are function algebras, Theorem E3 proves that all commutative subalgebras of 
A fundamental result due to Burnside (sometimes called the fundamental theorem of noncommutative algebra) asserts that every subalgebra 
if and only if 
is.
to 
A representation of 
and you are not expected to know the classification of subalgebras of 
,$
Therefore, we simply denote by 
the function defined by 
of an abstract algebra 
In the particular case of the endomorphism algebra 
is called a density operator if it satisfies 
Note that this condition is equivalent to
is a probability function in 
is a state on 
is faithful if and only if 
so that we can view 
. 
defined by 
, 
, is an injective algebra homomorphism.
in Theorem E8 is called the Wedderburn transform of 
Then, the scalar product on 
This means that the Wedderburn transform 
with respect to this Frobenius scalar product is defined.
and moreover 
for all 
:
: Endomorphism algebras.
: Convolution algebras.
: Class algebras.
: Polynomial algebras.
: Gelfand-Tsetlin algebras.
with operations defined pointwise. A basic family of functions are the elementary functions,
is a basis of 
of nonempty disjoint subsets of 
of 
are called its blocks. The set 
of partitions of 
if and only if 
can be obtained from 
is the subalgebra 
consisting of functions constant on the blocks of 
Note that 
is the complete collection of subalgebras of 
if and only if 
is said to separate points if, for every 
there exists a function 
of all subsets of 
is the ideal 
is the complete collection of ideals in 
if and only if 
the quotient algebra 
is isomorphic to the function algebra 
is called a probability distribution if 
for all 
, and 
for a unique probability distribution 
indexed by some finite set 
. The mapping 
is an algebra isomorphism called the Fourier transform. The “the” here is justified by the following.
exists, then it is unique up to relabeling the label set 
denoted 
and an antilinear, antimultiplicative, involution conjugation 
denoted 
Two further examples are 
with coordinatewise multiplication and conjugation, and 
with matrix multiplication and Hermitian conjugation. Both coincide with 
for 
Thus, we have an example of a commutative algebra of every dimension, and for algebras of square dimension bigger than one we also have a noncommutative example.
Two subclasses thereof are selfadjoint elements which are equal to their conjugate, 
, and unitary elements which satisfy 
Normal elements which are both selfadjoint and unitary are called reflections and they satisfy 
Selfadjoint elements which can be factored as 
are called nonnegative, and selfadjoint elements such that 
are called projections. Two projections which satisfy 
are said to be orthogonal.
with 
selfadjoint. Furthermore, this decomposition is unique and 
commute. 
has algebras as objects. Morphisms are linear transformations 
such that
are said to be isomorphic if and only if there exists a bijective homomorphism between them.
the set 
and is closed under multiplication and conjugation, i.e. it is a subalgebra of 
The set
However, 
is closed under multiplication and conjugation, and in fact for every 
we have 
for all 
and a unique maximal element of the poset of subalgebras of 
. There is also a unique minimal subalgebra containing both 
and it is denoted 
there is a unique minimal algebra 
containing 
, namely the intersection of all subalgebras containing it, and this is called the subalgebra generated by 
where 
if and only if 
can be viewed as a measure of how (non)commutative 
the corresponding centralizer is
then their centralizers are subalgebras such that 
there may not exist a commutative subalgebra containing it; this occurs for example if 
consists of a single non-normal element.
is contained in a commutative subalgebra of 
is commutative.
, but there may be many maximal commutative subalgebras 
having the property that they are not contained in any strictly larger commutative subalgebra. 
A scalar product so normalized is said to have the left-Frobenius property if
which has both the left-Frobenius property and the right-Frobenius property is called a Frobenius scalar product.
we can define a corresponding Hermitian form on 
satisfying 
and 
with equality if and only if 
which satisfies 
is the set of all operators which act diagonally on 
if and only if 
for each 
where 
is the orthogonal projection of 
A physicist would write this projection operator as 
and so that in Dirac notation the above decomposition becomes
. This is a particularly transparent instance of the Fourier transform. 
does there exist an orthonormal basis 
choose an arbitrary parameterization of the isomorphism classes of irreducible unitary representations of 
by the set 
of partitions of 
and for each 
choose an arbitrary representative 
of the corresponding isomorphism class. These choices being made, we define a graph 
called the branching graph of the tower of symmetric groups. The vertex set of 
for some 
The precise adjacency relation is the following: if 
and 
then 
is an edge of 
it contains an irreducible subspace ![V^\lambda[\mu]](https://s0.wp.com/latex.php?latex=V%5E%5Clambda%5B%5Cmu%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
isomorphic as a representation of 
and any ![V^\lambda = \bigoplus\limits_{\gamma \in \mathrm{Geo}(\mathrm{Par}_k,\lambda)} V^\lambda[\gamma],](https://s0.wp.com/latex.php?latex=V%5E%5Clambda+%3D+%5Cbigoplus%5Climits_%7B%5Cgamma+%5Cin+%5Cmathrm%7BGeo%7D%28%5Cmathrm%7BPar%7D_k%2C%5Clambda%29%7D+V%5E%5Clambda%5B%5Cgamma%5D%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
is the set of geodesics in 
to the specified point ![V^\lambda[\gamma]](https://s0.wp.com/latex.php?latex=V%5E%5Clambda%5B%5Cgamma%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
is a nonzero subspace of 
acts irreducibly. Note that while all of these subspaces 
the number of spaces in the above decomposition isomorphic to 
is 
where ![V^\lambda=\bigoplus\limits_{\gamma \in \mathrm{Geo}(1,\lambda)} V^\lambda[\gamma]](https://s0.wp.com/latex.php?latex=V%5E%5Clambda%3D%5Cbigoplus%5Climits_%7B%5Cgamma+%5Cin+%5Cmathrm%7BGeo%7D%281%2C%5Clambda%29%7D+V%5E%5Clambda%5B%5Cgamma%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
If we choose a unit vector 
from each Gelfand-Tsetlin line ![V^\lambda[\gamma],](https://s0.wp.com/latex.php?latex=V%5E%5Clambda%5B%5Cgamma%5D%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
we obtain an orthonormal basis of 
“the” Gelfand-Tsetlin basis of 
which really means that the basis is uniquely defined up to the phase ambiguity. In any case, this gives our first handle on the dimensions of the irreducible representations of the symmetric groups: we now know that 
and every 
let 
be the subalgebra of 
consisting of all elements 
belongs to the diagonal subalgebra 
of the GT-basis 
In other words, the image
under the Wedderburn isomorphism
-basis in every irreducible representations, the 
-block of 
of every irreducible representation 
of 
is the center of 
We had to put in some significant work to show that ![\mathcal{J}_n = \mathbb{C}[|J_1\rangle,\dots,|J_n\rangle]](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BJ%7D_n+%3D+%5Cmathbb%7BC%7D%5B%7CJ_1%5Crangle%2C%5Cdots%2C%7CJ_n%5Crangle%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
, which is shorthand for saying that the Jucys-Murphy specialization of the polynomial algebra ![\mathcal{P}_n=\mathbb{C}[x_1,\dots,x_n]](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BP%7D_n%3D%5Cmathbb%7BC%7D%5Bx_1%2C%5Cdots%2Cx_n%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
is a surjection onto 
This effort front loading is now starting to pay off. 
For this we use the fact, since we have already shown ![\mathcal{J}_n=\mathbb{C}[|J_1\rangle,\dots,|J_n\rangle],](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BJ%7D_n%3D%5Cmathbb%7BC%7D%5B%7CJ_1%5Crangle%2C%5Cdots%2C%7CJ_n%5Crangle%5D%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
it is sufficient to show that each JM-element 
belongs to the 
this reduces to showing that the sum 
of all transpositions in 
To demonstrate this, fix 
and let![V^\lambda = \bigoplus\limits_{\gamma \in \mathrm{Geo}(\mathrm{Par}_k,\lambda)} V^\lambda[\gamma]](https://s0.wp.com/latex.php?latex=V%5E%5Clambda+%3D+%5Cbigoplus%5Climits_%7B%5Cgamma+%5Cin+%5Cmathrm%7BGeo%7D%28%5Cmathrm%7BPar%7D_k%2C%5Clambda%29%7D+V%5E%5Clambda%5B%5Cgamma%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
belongs to one of the spaces 
acts as a scalar operator in each of these spaces by Schur’s Lemma.
using finite-dimensional Stone-Weierstrass and the fact that 
is isomorphic to the function algebra 
, where 
is the set of all geodesics from 
to 
of 
and in order to invoke Stone-Weirstrass to prove that 
we just have to show that 
are distinct geodesics 
in 
then there necessarily exists 
such that 
passes through 
passes through 
and 
In particular, 
are non-isomorphic irreducible representations of 
acts as a scalar operator in each of these irreducible representations: the eigenvalue of 
acting in 
for all 
this would force
be finite groups. In Lecture 12, we proved that restriction of irreducible representations of 
is multiplicity-free if and only if the centralizer 
is a commutative subalgebra of 
and 
By definition, the centralizer 
and it is clear that this condition is equivalent to
To understand those elements 
which have the above property, consider an equivalence relation on 
this is just the set of conjugacy classes in 
is an orthogonal basis of 
which is commutative by definition, and the above problem recovers the 202B fact that the center of the convolution algebra of a group has a basis consisting of indicator functions of conjugacy classes. At the other extreme, if 
is the subgroup consisting solely of the identity element, then 
which is commutative precisely when 
consists of selfadjoint elements, then 
by 
relabels the elements in the cycles of 
then it cannot relabel 
, hence in order for two permutations 
and 
in 
consisting of an integer 
corresponding to the length of the cycle containing 
consisting to the cycle-type of the remainder of the permutation. In particular, we get the dimension formula
is the number of partitions of 
. Furthermore, it is easy to see that if an 
contains a permutation 
, then it also contains 
, since the inverse of a permutation not only has the same cycle type, but actually the same elements within each cycle. Thus, the centralizer 
is commutative by Problem 13.2.
the set of elements in 
is commutative. Today, we will address this question in the case where 
is the convolution algebra of a finite group 
is the convolution algebra of a subgroup 
viewed as the subalgebra of 
be a set indexing isomorphism classes of irreducible unitary representations of 
let 
if and only if 
for each 
and every 
is commutative. If it were the case that 
this would be extremely easy, since then Schur’s Lemma would force 
So, we will have to understand what conditions would cause 
be a set parameterizing isomorphism classes of irreducible unitary representations of ![V^\lambda =\bigoplus\limits_{\mu \in \Lambda(H)} V^\lambda[\mu],](https://s0.wp.com/latex.php?latex=V%5E%5Clambda+%3D%5Cbigoplus%5Climits_%7B%5Cmu+%5Cin+%5CLambda%28H%29%7D+V%5E%5Clambda%5B%5Cmu%5D%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
![V^\lambda[\nu]](https://s0.wp.com/latex.php?latex=V%5E%5Clambda%5B%5Cnu%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
are isomorphic unitary representations of 
In finer detail, each ![V^\lambda[\mu]=W^\mu_1 \oplus \dots \oplus W^\mu_{m_{\lambda\mu}},](https://s0.wp.com/latex.php?latex=V%5E%5Clambda%5B%5Cmu%5D%3DW%5E%5Cmu_1+%5Coplus+%5Cdots+%5Coplus+W%5E%5Cmu_%7Bm_%7B%5Clambda%5Cmu%7D%7D%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
and 
are isomorphic irreducible unitary representations of 
is a positive integer.
, then ![T \in \mathrm{End}_H(V^\lambda[\mu])](https://s0.wp.com/latex.php?latex=T+%5Cin+%5Cmathrm%7BEnd%7D_H%28V%5E%5Clambda%5B%5Cmu%5D%29&bg=FFFFFF&fg=000&s=0&c=20201002)
for a fixed 
. Each such operator may be visualized as a block matrix
Now, since 
Schur’s Lemma tells us that 
is one-dimensional, so in fact each block 
can be identified with a scalar 
and 
matrix of complex numbers. We thus have an algebra isomorphism
for each 
We have thus established the following theorem.
is a surjection of the algebra ![\mathcal{S}_n=\mathbb{C}[x_1,\dots,x_n]^{S_n}](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BS%7D_n%3D%5Cmathbb%7BC%7D%5Bx_1%2C%5Cdots%2Cx_n%5D%5E%7BS_n%7D&bg=FFFFFF&fg=000&s=0&c=20201002)
of symmetric polynomials onto the center 
of the convolution algebra of the symmetric group 
there exists a symmetric polynomial 
such that 
where 
on the Jucys-Murphy elements
of the Cayley graph 
of 
of transpositions, we can say that there exists a (not necessarily symmetric) polynomial ![g \in \mathcal{P}_n=\mathbb{C}[x_1,\dots,x_n]](https://s0.wp.com/latex.php?latex=g+%5Cin+%5Cmathcal%7BP%7D_n%3D%5Cmathbb%7BC%7D%5Bx_1%2C%5Cdots%2Cx_n%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
such that 
;
![\mathcal{Z}_n=\mathbb{C}[|J_1\rangle,\dots,|J_n\rangle]^{S_n}=\mathbb{C}[|L_0\rangle,\dots,|L_{n-1}\rangle],](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BZ%7D_n%3D%5Cmathbb%7BC%7D%5B%7CJ_1%5Crangle%2C%5Cdots%2C%7CJ_n%5Crangle%5D%5E%7BS_n%7D%3D%5Cmathbb%7BC%7D%5B%7CL_0%5Crangle%2C%5Cdots%2C%7CL_%7Bn-1%7D%5Crangle%5D%2C&bg=FFFFFF&fg=000&s=0&c=20201002)
,
for the variables of symmetric polynomials in 
for the variables of arbitrary polynomials in 
of 
Our original proof of the fact that the JM-elements commute was simply to point out that they lie in the commutative algebra 
surjects 
onto 
the natural guess is that it surjects 
onto ![\mathcal{P}_\mathbb{C}[x_1,\dots,x_n]](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BP%7D_%5Cmathbb%7BC%7D%5Bx_1%2C%5Cdots%2Cx_n%5D&bg=FFFFFF&fg=000&s=0&c=20201002)
onto 
