Stable Schur functions from bialternants

For a partition $\lambda$ and an integer $N\ge \mathrm{\ell }(\lambda )$, pad $\lambda$ with zero parts to length $N$ and set ${\delta }_N=(N-1,N-2,\dots ,0)$. Define the alternating polynomial $a_{\lambda +{\delta }_N}(x_1,\dots ,x_N):=\det (x_i^{{\lambda }_j+N-j}{)}_{1\le i,j\le N},$ and define $a_{{\delta }_N}$ by the same formula with $\lambda =\mathrm{\varnothing }$. The finite-rank Schur polynomial is $s_{\lambda }(x_1,\dots ,x_N):=\frac{a_{\lambda +{\delta }_N}(x_1,\dots ,x_N)}{a_{{\delta }_N}(x_1,\dots ,x_N)}\mathrm{.}$ For $0\le N<\mathrm{\ell }(\lambda )$, define $s_{\lambda }(x_1,\dots ,x_N):=0$. Thus a component is specified at every rank, including rank zero; empty determinants have value $1$.

Well-definedness and stability. The exponents ${\lambda }_j+N-j$ are strictly decreasing, so the numerator is alternating. Setting $x_i=x_j$ makes it zero; the factor theorem therefore gives divisibility by each $x_i-x_j$ in $\mathbb{Z}[x_1,\dots ,x_N]$. These pairwise nonassociate prime factors therefore have product $a_{{\delta }_N}$ dividing the numerator, so the quotient is an integral polynomial. Since numerator and denominator both change by the sign of a variable permutation, their quotient is symmetric. Its degree is $\mathrm{\mid }\lambda \mathrm{\mid }$.

If $N+1>\mathrm{\ell }(\lambda )$, setting $x_{N+1}=0$ in the rank-$(N+1)$ numerator and denominator expands each determinant along its last row; both resulting minors have the common factor $x_1\cdots x_N$, and after cancelling it the quotient is exactly the rank-$N$ quotient. At the remaining boundary $N+1=\mathrm{\ell }(\lambda )>0$, every exponent in the rank-$(N+1)$ numerator is positive, so its last row becomes zero when $x_{N+1}=0$. The denominator specializes to $(x_1\cdots x_N)a_{{\delta }_N}$, a nonzero polynomial (equal to $1$ when $N=0$). Specializing the polynomial identity $a_{\lambda +{\delta }_{N+1}}=a_{{\delta }_{N+1}}{s}_{\lambda }$ and cancelling this nonzero polynomial shows that the specialized Schur polynomial is zero, as required by the rank-$N$ definition. Below this boundary both components are zero. Hence these polynomials form a compatible sequence in the inverse limit The stable graded ring of symmetric functions, defining $s_{\lambda }\in {\mathrm{\Lambda }}^{\mathrm{\mid }\lambda \mathrm{\mid }}$. With the empty determinant equal to $1$, $s_{\mathrm{\varnothing }}=1$. The partition length and padding convention is that of Partitions, English diagrams, and conjugation.

At the minimal rank $N=1$, the one-box partition gives $s_{(1)}(x_1)=x_1$ directly from the quotient; its stable sequence is nonzero.