Algebraic combinatorics Lecture 9: symmetric group representations and isomorphism of symmetric functions and class functions

Since I am not a superhuman, my blog lengths will significantly reduce in size from now on. Let {L(G)} be the space of functions on a finite group {G}. We can define the usual inner product on it, and the convolution makes it into a group algebra; one can associate each function with an element of the group algebra in the obvious way. The center of {L(G)} consists of class functions, as testified by the indicator functions of single elements. Thus a basis of the center is given by {\delta_C}, where {C} ranges over conjugacy classes. For class functions on {S_n},

\displaystyle  \langle f, h \rangle = \sum_\lambda \frac{f(\lambda) h(\lambda)}{z_\lambda}. \ \ \ \ \ (17)

Definition 9 Let {CL = \bigoplus_{n=0}^\infty CL_n} to be the graded algebra of class functions, where multiplication is defined by

\displaystyle  f \circ g = \rm{IND}_{S_n \times S_m}^{S_{n+m}}(f \times g) \in CL_{n+m}. \ \ \ \ \ (18)

Further let {\rm{ch}: CL \rightarrow \Lambda} called the characteristic map be defined by

\displaystyle  \rm{ch}(f) = \frac{1}{n!} \sum_{\omega \in S_n} f(\omega) p_{\rho(\omega)} = \sum_{\lambda\vdash n} f(\lambda) p_\lambda / z_\lambda. \ \ \ \ \ (19)

Proposition 10 Ch is an isometry.

The induced map {\rm{IND}_{S_n \times S_m}^{S_{n+m}}(f)} is defined by the Frobenius relation:

\displaystyle  \langle \rm{IND}_{S_n \times S_m}^{S_{n+m}}(f), \chi \rangle_G = \langle f, \rm{Res}_H^G \chi\rangle_G \ \ \ \ \ (20)

or equivalently,

\displaystyle  \rm{IND}_{S_n \times S_m}^{S_{n+m}}(f)(s) = \sum_{i=0}^k \bar{f}(t_i^{-1} s t_i), \ \ \ \ \ (21)

where {t_i} are representative of right cosets of {H} in {G}, and {\bar{f}} equals f on H and 0 o.w.
The proposition above is easy, next we show

Proposition 11 Ch is an algebra isomorphism.

Proof: One can write

\displaystyle  Ch(f) = \langle f, \psi \rangle \ \ \ \ \ (22)

where {\psi: S_n \rightarrow \Lambda^n} given by {\psi(\omega) = p_{\rho(\omega)}}. Note {\psi} is a function on {S_n} with value in the space of symmetric functions. This coupled with Frobenius reciprocity easily prove the assertion. \Box

Need some representation theory of {S_n}. For a G-set X (G acts transitively), one has the permutation reprsentation on it. For set partitions of {[n]} given by a partition {\lambda}, called partial rankings, {S_N} acts on them in the natural way, i.e., permuting the labels. This representation is called Specht module.
Fact: {m^\lambda = \rm{IND}_{S_\lambda}^{S_n} (1)}, where {S_\lambda = S_{\lambda_1} \times \ldots \times S_{\lambda_k} \subset S_n}. But these representations are reducible, for instance:

\displaystyle  m^{n-1,1} = S^n \oplus S^{n-1,1}. \ \ \ \ \ (23)

Let {\eta^\lambda} be the character of {m^\lambda}.

Proposition 12

\displaystyle  \rm{ch}(\eta^\lambda) = h_\lambda = h_{\lambda_1} \ldots h_{\lambda_l} \ \ \ \ \ (24)

Proof: First we establish it for {f = 1 = \delta_{S_n}}. So we basically need {Ch(f) = \sum_\lambda \frac{p_\lambda}{z_\lambda} \overset{?}{=} h_n}. Recall {\prod_{i,j} (1-x_i y_j)^{-1} = \sum_\lambda \frac{p_\lambda(x) p_\lambda(y)}{z_\lambda}}. Setting {y_1 = 1}, {y_2 = \ldots =0}, then {p_\lambda(y) = t^{|\lambda|}}. Then

\displaystyle  \prod_i (1-tx_i)^{-1} = \sum_\lambda \frac{p_\lambda(x)}{z_\lambda} t^{|\lambda|}\\ = \sum_n t^n \sum_{\lambda \vdash n} \frac{p_\lambda(x)}{z_\lambda} \ \ \ \ \ (25)

But {[t^n] \prod_i (1-x_i t)^{-1} = h_n}.

\displaystyle  Ch(\eta^\lambda) = Ch(\delta_{S_{\lambda_1}} \circ \ldots \circ \delta_{S_{\lambda_l}}) = h_{\lambda_1} h_{\lambda_2} \ldots h_{\lambda_l} = h_\lambda \ \ \ \ \ (26)


Next we find linear combinations of {\{\eta_\lambda\}} giving an irreducible character. Recall Munagen-Nakayama:

\displaystyle  p_\mu = \sum_{\lambda \vdash n} \chi^\lambda(\mu) s_\lambda\\ s_\mu = \sum_{\lambda \vdash n} \frac{\chi^\lambda(\mu)}{z_\lambda} p_\lambda. \ \ \ \ \ (27)

{\chi^\lambda(\mu)} is the number of tableaux of shape {\lambda} and type {\mu}. Define the following linear combination of products of {\eta_\lambda}‘s:

\displaystyle  \psi^\lambda = \det (\eta^{\lambda_i + j -i}) \ \ \ \ \ (28)

in analogy of Jacobi-Trudi formula for Schur functions. Then using isometry and algebra isomorphism of {\rm{ch}}, we see

\displaystyle  \langle \psi^\lambda, \psi^\nu \rangle = \delta_{\lambda \nu}. \ \ \ \ \ (29)

Since the {\eta^k}‘s are characters of possibly reducible representations, which in turn are {{\mathbb Z}}-linear combinations of irreps, we know that {\pm \psi^\lambda}‘s must be irreducible characters. And since {\psi^\lambda(1) = } the number of SSYT’s which is positive, we know that {\psi^\lambda} is the irreducible characters (they evaluate to a positive trace on identity) and can be identified with {\chi^\lambda}.


About aquazorcarson

math PhD at Stanford, studying probability
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