$\eta$-value of a partition and its meaning

Question

The $\eta$-value of an integer partition $\lambda = \big( \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 0 \big)$ is defined as

\begin{equation} \eta \big( \lambda \big) \ := \ \sum_{i=1}^k \, (i-1) \, \lambda_i \end{equation}

It makes an appearance in a $q$-version of the hook-length formula for a special value of the Schur function $s_\lambda$, namely

\begin{equation} s_\lambda \big(1, q, q^2, \dots \big) \ = \ q^{\eta(\lambda)} \, \prod_{\stackrel{\scriptstyle \text{boxes}}{b \, \in \, \lambda}} \, \Big(1 \, - \, q^{\, \text{hook}\, (b)} \Big)^{-1} \end{equation}

where $q$ is a variable, $b$ is a box in the young diagram of $\lambda$, and $\text{hook}(b)$ is the hook-length of the box $b$.

Question: Is there a natural interpretation of $\eta(\lambda)$ as class-function on the symmetric group $S_n$ where $n = |\lambda|$ and where $\lambda$ labels the conjugacy class in $S_n$ consisting of permutations of cycle-type $\lambda$ ?

ines.

Answer 1

Dear Ines,

Let me supplement my first response with a related (possible) answer to your question which makes use of the expansion of power-symmetric functions in terms of Schur functions (an identity which you and I have discussed in other threads of this forum).

Given a partition $\mu= \big(\mu_1 \geq \dots \geq \mu_k > 0 \big)$ of $n$ with $k$ parts, you will recall that the corresponding power symmetric function $p_\mu$ is defined as the product $p_{\mu_1} \cdots \, p_{\mu_k}$ of elementary power-symmetric functions $p_{\mu_1}, \dots, p_{\mu_k}$ where

\begin{equation} p_r \ = \ \sum_{i \geq 1} \, x_i^r \end{equation}

for each positive integer $r$. The expansion of a power-symmetric function in terms of Schur functions is

\begin{equation} p_\mu \ = \ \sum_{|\lambda| \, = \, n} \, \chi^\lambda_\mu \, s_\lambda \qquad \qquad (\dagger) \end{equation}

where the sum is taken over all partitions $\lambda$ of $n$ and $\chi^\lambda_\mu$ is the value of the irreducible character $\chi^\lambda$ of the symmetric group $S_n$ evaluated on the conjugacy class $C_\mu$. We'll begin by specialising the left and right hand sides of equation $(\dagger)$ using the values $x_i = q^{\, i}$. The power-symmetric case is a simple calculation involving geometric series and one checks that

\begin{equation} p_\mu \big(1, q, q^2, \dots \big) \ = \ \big(1-q \big)^{-k} \cdot \big[ \, \mu_1 \big]^{-1} \cdots \big[ \, \mu_k \big]^{-1} \qquad \qquad (*) \end{equation}

where $\displaystyle [r] = {\, \, \, 1 - q^{\, r} \over {1 -q}} $ is the combinatorist's $q$-number of a positive integer $r$. The specialisation of the Schur function is the celebrated gem (of Stanley ?) with which you began the discussion, namely

\begin{equation} \begin{array}{ll} \displaystyle s_\lambda \big(1, q, q^2, \dots \big) &\displaystyle = \ q^{\eta(\lambda)} \, \prod_{\stackrel{\scriptstyle \text{boxes}}{b \, \in \, \lambda}} \, \Big(1 \, - \, q^{\, \text{hook}\, (b)} \Big)^{-1} \\ \\ &\displaystyle = \ q^{\, \eta(\lambda) \, + \, H(\lambda)} \, \big(1 -q \big)^{-n} \, \prod_{\stackrel{\scriptstyle \text{boxes}}{b \, \in \, \lambda}} \, \big[\!\big[ \, \text{hook}(b) \, \big]\!\big]^{-1} \\ \\ &\displaystyle = \ q^{ {1 \over 2} \big( \eta(\lambda) \, - \,\eta(\lambda^{\text{op}}) \big)} \, \big(1 -q \big)^{-n} \, \prod_{\stackrel{\scriptstyle \text{boxes}}{b \, \in \, \lambda}} \, \big[\!\big[ \, \text{hook}(b) \, \big]\!\big]^{-1} \qquad \qquad (**) \end{array} \end{equation}

where $\displaystyle [\![ \, r \, ]\!] = {q^{\scriptstyle {1 \over 2}r} - q^{ \scriptstyle -{1 \over 2}r} \over {q^{ \scriptstyle {1 \over 2}}} - q^{\scriptstyle -{1 \over 2}}}$ is the physicist's $q$-number of a positive integer $r$, where $\text{hook}(b)$ is the hook-length of a box in the young diagram of $\lambda$, and where

\begin{equation} \begin{array}{ll} \displaystyle H(\lambda) \ &\displaystyle = \ {n \over 2} \, - \, {1 \over 2} \, \sum_{\stackrel{\scriptstyle \text{boxes}}{b \, \in \, \lambda}} \text{hook}(b) \\ \\ &\displaystyle = \ - {1 \over 2} \Big( \eta \big(\lambda \big) \, + \, \eta \big( \lambda^{\text{op}} \big) \Big) \end{array} \end{equation}

The physicist's $q$ numbers are more convenient in this present context because $[\![ \, r \, ]\!] = r$ and $\displaystyle {d \over {dq}} \, [\![ \, r \, ]\!] = 0$ at $q=1$. Inserting formulae $(*)$ and $(**)$ into equation $(\dagger)$ we obtain

\begin{equation} \big(1-q \big)^{n-k} \cdot \big[ \, \mu_1 \big]^{-1} \cdots \big[ \, \mu_k \big]^{-1} \ = \ \sum_{|\lambda| \, = \, n} \, \chi^\lambda_\mu \ q^{ {1 \over 2} \big( \eta(\lambda) \, - \,\eta(\lambda^{\text{op}}) \big)} \, \prod_{\stackrel{\scriptstyle \text{boxes}}{b \, \in \, \lambda}} \, \big[\!\big[ \, \text{hook}(b) \, \big]\!\big]^{-1} \end{equation}

Upon differentiating by $q$, specialising $q=1$, and invoking the (classical) hook-length formula we obtain

\begin{equation} \begin{array}{ll} \displaystyle - \delta_{k, n-1} \cdot \mu_1^{-1} \cdots \mu_k^{-1} &\displaystyle = \ \sum_{|\lambda| \, = \, n} \, \chi^\lambda_\mu \ {1 \over 2} \Big( \eta(\lambda) \, - \,\eta(\lambda^{\text{op}}) \Big) \, \prod_{\stackrel{\scriptstyle \text{boxes}}{b \, \in \, \lambda}} \, \text{hook}^{-1}(b) \\ \\ &\displaystyle = \ \sum_{|\lambda| \, = \, n} \, \chi^\lambda_\mu \, \ {\chi^\lambda_{1^n} \over {2n!}} \,\Big(\eta(\lambda) \, - \,\eta(\lambda^{\text{op}}) \Big) \end{array} \end{equation}

where $1^n$ is the partition corresponding to the conjugacy class of the identity element in $S_n$. Define

\begin{equation} F(\lambda) \ := \ {\chi^\lambda_{1^n} \over {2n!}} \,\Big(\eta(\lambda) \, - \,\eta(\lambda^{\text{op}}) \Big) \end{equation}

In view of Schur's second character orthogonality relations it follows that $F(\lambda)$ is proportional to $\chi^\lambda_\varrho$ where $\varrho = 21^{n-1}$ is the partition corresponding to the conjugacy class $T$ consisting of all transpositions in $S_n$. Check that in fact

\begin{equation} F(\lambda) \ = \ -{1 \over 2} \,{|T| \over {n!}} \, \chi^\lambda_\varrho \end{equation}

and so arrive at the following character-theoretic account of the $\eta$-value of a partition, namely:

\begin{equation} \eta\big( \lambda \big) \, - \, \eta\big( \lambda^{\text{op}} \big) \ = \ -|T| \, {\chi^\lambda_\varrho \over {\chi^\lambda_{1^n} }} \end{equation}

yours, a. leverkühn

Answer 2

Here's a story that might please you: Let $T$ be the set of all transpositions in $S_n$ and define a corresponding discrete Laplace operator $\Delta$ by

\begin{equation} \Delta f ( \sigma ) \ = \ \sum_{\tau \in T} \, f ( \sigma ) - f ( \tau \cdot \sigma ) \end{equation}

for any complex-valued function $f: S_n \longrightarrow \Bbb{C}$ and permutation $\sigma \in S_n$. It's not hard to check that $\Delta f$ is a class funtion whenever $f$ is. For $\sigma \in S_n$ consider its transposition length $l(\sigma)$, namely the minimal number of transpositions required to factorise $\sigma$. The mapping $\sigma \mapsto l(\sigma)$ is evidently a class function and furthermore $l(\sigma) = n-k$ where $k$ is number of parts of the partition $\lambda$ which encodes the cycle type of $\sigma$. If my calculations are correct then

\begin{equation} \Delta l (\sigma) \ = \ 2 \eta\big( \lambda^\text{op} \big) \, - \, \binom{n}2 \end{equation}

where $\lambda$ is the partition encoding the cycle type of $\sigma$ and $\lambda^\text{op}$ is the conjugate partition. yours, A. Leverkühn