Summary: Symmetric functions transition table

Over the last few weeks I’ve been writing about several little gemstones that I have seen in symmetric function theory. But one of the main overarching beauties of the entire area is that there are at least five natural bases with which to express any symmetric functions: the monomial ($m_\lambda$), elementary ($e_\lambda$), power sum ($p_\lambda$), complete homogeneous ($h_\lambda$), and Schur ($s_\lambda$) bases. As a quick reminder, here is an example of each, in three variables $x,y,z$:

$m_{(3,2,2)}=x^3y^2z^2+y^3x^2z^2+z^3y^2x^2$

$e_{(3,2,2)}=e_3e_2e_2=xyz(xy+yz+zx)^2$

$p_{(3,2,2)}=p_3p_2p_2=(x^3+y^3+z^3)(x^2+y^2+z^2)^2$

$h_{(2,1)}=h_2h_1=(x^2+y^2+z^2+xy+yz+zx)(x+y+z)$

$s_{(3,1)}=m_{(3,1)}+m_{(2,2)}+2m_{(2,1,1)}$

Since we can usually transition between the bases fairly easily, this gives us lots of flexibility in attacking problems involving symmetric functions; it’s sometimes just a matter of picking the right basis.

So, to wrap up my recent streak on symmetric function theory, I’ve posted below a list of rules for transitioning between the bases. (The only ones I have not mentioned is how to take a polynomial expressed in the monomial symmetric functions $m_\lambda$ in terms of the others; this is rarely needed and also rather difficult.)

Elementary to monomial: \[e_\lambda=\sum M_{\lambda\mu} m_\mu\] where $M_{\lambda\mu}$ is the number of $0,1$-matrices with row sums $\lambda_i$ and column sums $\mu_j$.

Elementary to homogeneous: \[e_n=\det \left(\begin{array}{cccccc} h_1 & 1 & 0 & 0 &\cdots & 0 \\ h_2 & h_1 & 1 & 0 & \cdots & 0 \\ h_3 & h_2 & h_1 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ h_{n-1} & h_{n-2} & h_{n-3} & h_{n-4} & \ddots & 1 \\ h_n & h_{n-1} & h_{n-2} & h_{n-3} & \cdots & h_1 \end{array}\right)\]

Elementary to power sum: \[e_n=\frac{1}{n!}\det\left(\begin{array}{cccccc} p_1 & 1 & 0 & 0 &\cdots & 0 \\ p_2 & p_1 & 2 & 0 & \cdots & 0 \\ p_3 & p_2 & p_1 & 3 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ p_{n-1} & p_{n-2} & p_{n-3} & p_{n-4} & \ddots & n-1 \\ p_n & p_{n-1} & p_{n-2} & p_{n-3} & \cdots & p_1 \end{array}\right)\]

Elementary to Schur: \[e_\lambda=\sum_{\mu} K_{\mu’\lambda}s_\mu\] where $K_{\lambda\mu}$ is the number of semistandard Young tableau of shape $\lambda$ and content $\mu$.

Homogeneous to monomial: \[h_\lambda=\sum N_{\lambda\mu} m_\mu\]

where $N_{\lambda\mu}$ is the number of matrices with nonnegative integer entries with row sums $\lambda_i$ and column sums $\mu_j$.

Homogeneous to elementary: \[h_n=\det\left(\begin{array}{cccccc} e_1 & 1 & 0 & 0 &\cdots & 0 \\ e_2 & e_1 & 1 & 0 & \cdots & 0 \\ e_3 & e_2 & e_1 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ e_{n-1} & e_{n-2} & e_{n-3} & e_{n-4} & \ddots & 1 \\ e_n & e_{n-1} & e_{n-2} & e_{n-3} & \cdots & e_1 \end{array}\right)\]

Homogeneous to power sum: \[h_n=\frac{1}{n!}\det\left(\begin{array}{cccccc} p_1 & -1 & 0 & 0 &\cdots & 0 \\ p_2 & p_1 & -2 & 0 & \cdots & 0 \\ p_3 & p_2 & p_1 & -3 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ p_{n-1} & p_{n-2} & p_{n-3} & p_{n-4} & \ddots & -(n-1) \\ p_n & p_{n-1} & p_{n-2} & p_{n-3} & \cdots & p_1 \end{array}\right)\]

Homogeneous to Schur: \[h_{\lambda}=\sum_\mu K_{\mu\lambda}s_\mu\]

Power sum to monomial: \[p_\lambda=\sum_{\mu} R_{\lambda\mu}m_\mu\] where $R_{\lambda\mu}$ is the number of ways of sorting the parts of $\lambda$ into a number of ordered blocks in such a way that the sum of the parts in the $j$th block is $\mu_j$.

Power sum to elementary: Newton-Gerard identities: \[p_n=\det\left(\begin{array}{cccccc} e_1 & 1 & 0 & 0 &\cdots & 0 \\ 2e_2 & e_1 & 1 & 0 & \cdots & 0 \\ 3e_3 & e_2 & e_1 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ (n-1)e_{n-1} & e_{n-2} & e_{n-3} & e_{n-4} & \ddots & 1 \\ ne_n & e_{n-1} & e_{n-2} & e_{n-3} & \cdots & e_1 \end{array}\right)\]

Power sum to homogeneous: \[p_n=(-1)^{n-1}\det\left(\begin{array}{cccccc} h_1 & 1 & 0 & 0 &\cdots & 0 \\ 2h_2 & h_1 & 1 & 0 & \cdots & 0 \\ 3h_3 & h_2 & h_1 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ (n-1)h_{n-1} & h_{n-2} & h_{n-3} & h_{n-4} & \ddots & 1 \\ nh_n & h_{n-1} & h_{n-2} & h_{n-3} & \cdots & h_1 \end{array}\right)\]

Power sum to Schur: Let $\chi^\lambda$ be the $\lambda$th character of the symmetric group $S_n$ where $n=|\lambda|$, and write $\chi^\lambda(\mu)$ to denote the value of $\chi^{\lambda}$ at any permutation with cycle type $\mu$. Then for any partition $\mu\vdash n$, we have: \[p_\mu=\sum_{\lambda\vdash n} \chi^\lambda(\mu) s_\lambda\] Alternatively: \[p_n=s_{(n)}-s_{(n-1,1)}+s_{(n-2,1,1)}-s_{(n-3,1,1,1)}+\cdots+(-1)^{n}s_{(1,1,\ldots,1)}\]

Schur to monomial: \[s_{\lambda}=\sum_{\mu\vdash |\lambda|} K_{\lambda \mu}m_\mu\]

Schur to elementary: (Dual Jacobi-Trudi Identity.) \[s_{\lambda/\mu} = \det \left(e_{\lambda’_i-\mu’_j-i+j}\right)_{i,j=1}^n\]

Schur to homogeneous: (Jacobi-Trudi Identity.) \[s_{\lambda/\mu} = \det \left(h_{\lambda_i-\mu_j-i+j}\right)_{i,j=1}^n\]

Schur to power sum: \[s_\lambda=\sum_{\nu\vdash |\lambda|} z_\nu^{-1} \chi^{\lambda}(\nu) p_\nu\]