The symmetric functions catalog

#Back stable Schubert polynomials

Let \(\symS_{\setZ}\) be the permutations of \(\setZ\) with finite support, and let \(\gamma(s_i)=s_{i+1}\) and \(\gamma(x_i)=x_{i+1}.\) For \(w\in\symS_\infty,\) the back stable Schubert polynomial is \[\overleftarrow{\schubert}_w = \lim_{m\to\infty}\gamma^{-m}\schubert_{1^m\times w}.\] The definition extends to every \(w\in\symS_{\setZ}\) and is independent of the finite interval used to represent \(w\) [Sec. 2.3, Rod26].

#The back stable Schubert ring

Let \(\Lambda^-\) be the symmetric functions in \(\dotsc,x_{-1},x_0.\) The back stable Schubert polynomials form a basis of \[\mathcal R = \Lambda^-\otimes_{\setQ}\setQ[x_i:i\in\setZ]\] [Thm. 3.5, LLS18]. Killing the nonpositive variables recovers the ordinary Schubert polynomial. Back stable Schubert polynomials also have bumpless pipe-dream formulas; double and enriched versions appear among the Schubert polynomial variations.

#Standard monomial bases

The products \[e_\alpha=\prod_{i\geq1}e_{\alpha_i}(x_1,\dotsc,x_i), \qquad 0\leq\alpha_i\leq i,\] are the standard elementary monomials ; they form a basis of the ordinary polynomial ring [FGP97, Win98].

For a finitely supported tuple \(\alpha=(\alpha_i)_{i\in\setZ},\) Rodriguez defines the back stable standard elementary monomial and its complete homogeneous analogue by \[\overleftarrow{e}_\alpha =\prod_{i\in\setZ}e_{\alpha_i}(\dotsc,x_{i-1},x_i), \qquad \overleftarrow{h}_\alpha =\prod_{i\in\setZ}h_{\alpha_i}(\dotsc,x_{i-1},x_i).\] Each family is a basis of \(\mathcal R\) [Thm. 3.3 and Cor. 4.5, Rod26]. Write \[\overleftarrow{\schubert}_w = \sum_\alpha \overleftarrow{\kappa}^w_\alpha\,\overleftarrow{e}_\alpha.\] The coefficients \(\overleftarrow{\kappa}^w_\alpha\) can have either sign.

#Stabilization

The coefficients \(\overleftarrow{\kappa}^w_\alpha\) determine the standard elementary expansion of every sufficiently far shift \(\schubert_{\gamma^m(w)}\) [Thm. 3.7, Rod26]. Thus one back stable expansion records the eventual behavior of the corresponding ordinary Schubert expansions.

#Dynkin reversal

The involution \[\widetilde\omega(e_r)=h_r, \qquad \widetilde\omega(x_i)=-x_{1-i}\] exchanges the two standard monomial bases. If \(w=s_{a_1}\dotsm s_{a_\ell},\) its Dynkin reversal is \(\widehat w=s_{-a_1}\dotsm s_{-a_\ell},\) and \[\widetilde\omega(\overleftarrow{\schubert}_w) = \overleftarrow{\schubert}_{\widehat w}\] [Thms. 4.3–4.6, Rod26].

#Stanley symmetric functions

Let \(\operatorname{par}(\alpha)\) be the partition obtained by sorting the nonzero entries of \(\alpha.\) The orbit sums of the back stable coefficients recover the elementary coefficients of the Stanley symmetric function: \[\sum_{\operatorname{par}(\alpha)=\lambda} \overleftarrow{\kappa}^w_\alpha = [e_\lambda]F_w\] [Thm. 5.4, Rod26].

#Shifted specialization

For \(w\in\symS_\infty\) and \(t\geq0,\) Rodriguez’s shifted specialization satisfies \[\mathcal T_w(t) = \schubert_{1^t\times w}(1) = |\operatorname{RP}(1^t\times w)|,\] so it counts reduced pipe dreams after adjoining \(t\) fixed points [Def. 6.1 and Prop. 6.3, Rod26].

Bibliography

1. [FGP97]Sergey Fomin, Sergei Gelfand and Alexander Postnikov. Quantum Schubert polynomials. Journal of the American Mathematical Society, 10(03):565–597, July 1997.
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   @article{FominGelfandPostnikov1997,
     doi = {10.1090/s0894-0347-97-00237-3},
     url2 = {https://doi.org/10.1090/s0894-0347-97-00237-3},
     year  = {1997},
     month = jul,
     publisher = {American Mathematical Society ({AMS})},
     volume = {10},
     number = {03},
     pages = {565--597},
     title = {Quantum {S}chubert polynomials},
     author = {Sergey Fomin and Sergei Gelfand and Alexander Postnikov},
     journal = {Journal of the American Mathematical Society}
   }
   

2. [LLS18]Thomas Lam, Seung Jin Lee and Mark Shimozono. Back stable Schubert calculus. arXiv:1806.11233, 2018.
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   @article{LamLeeShimozono2018x,
   Author = {Thomas Lam and Seung Jin Lee and Mark Shimozono},
   Title = {Back stable {S}chubert calculus},
   Year = {2018},
   journal = {arXiv e-prints},
   Eprint = {1806.11233}
   }
   

3. [Rod26]Carlos Rodriguez. Back Stable Standard Elementary Monomials. arXiv:2609.25445v1, 2026.
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   @article{Rodriguez2026x,
     author = {Carlos Rodriguez},
     title = {Back {S}table {S}tandard {E}lementary {M}onomials},
     year = {2026},
     eprint = {2609.25445v1},
     url = {https://arxiv.org/abs/2609.25445v1},
     journal = {arXiv e-prints}
   }
   

4. [Win98]Rudolf Winkel. On the expansion of Schur and Schubert polynomials into standard elementary monomials. Advances in Mathematics, 136(2):224–250, June 1998.
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   @article{Winkel1998,
     doi = {10.1006/aima.1998.1730},
     url2 = {https://doi.org/10.1006/aima.1998.1730},
     year  = {1998},
     month = jun,
     publisher = {Elsevier {BV}},
     volume = {136},
     number = {2},
     pages = {224--250},
     author = {Rudolf Winkel},
     title = {On the Expansion of {S}chur and {S}chubert Polynomials into Standard Elementary Monomials},
     journal = {Advances in Mathematics}
   }