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Darondeau–Pragacz formulas in complex cobordism

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Abstract

In this paper, we generalize the push-forward (Gysin) formulas for flag bundles in ordinary cohomology theory, which are due to Darondeau–Pragacz, to the complex cobordism theory. Then, we introduce the universal quadratic Schur functions, which are a generalization of the (ordinary) quadratic Schur functions introduced by Darondeau–Pragacz, and establish some Gysin formulas for the universal quadratic Schur functions as an application of our Gysin formulas.

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Notes

  1. D–P formulas for short.

  2. For further applications of Gysin formulas in complex cobordism and techniques of generating functions, readers are referred to a companion paper [20].

  3. We adopted the terminology used in Darondeau–Pragacz [8, §1.2].

  4. An analogous determinantal formula for the Grothendieck polynomials was recently obtained by Hudson–Ikeda–Matsumura–Naruse [13, Theorem 3.13].

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Acknowledgements

We would like to thank Piotr Pragacz and Tomoo Matsumura for valuable discussions, and Eric Marberg for pointing out an ambiguity in our treatment of the residue symbol \(\underset{t = 0}{\mathrm {Res}'}\) (see Sect. 3.1.2 and Sect. 5.1). Thanks are also due to the anonymous referee whose careful reading and useful comments greatly improved our previous manuscript. The first author is partially supported by JSPS KAKENHI Grant number JP18K03303, Japan Society for the Promotion of Science. The second author is partially supported by JSPS KAKENHI Grant number JP16H03921, Japan Society for the Promotion of Science.

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Authors and Affiliations

  1. Graduate School of Education, Okayama University, Okayama, 700-8530, Japan

    Masaki Nakagawa

  2. Graduate School of Education, University of Yamanashi, Kofu, 400-8510, Japan

    Hiroshi Naruse

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  1. Masaki Nakagawa
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  2. Hiroshi Naruse
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Correspondence to Masaki Nakagawa.

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Communicated by Jean-Yves Welschinger.

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Appendix

Appendix

1.1 Quillen’s Residue Formula

As we mentioned in Sect. 3.1.2, we proceed with the calculation of Quillen’s residue formula, i.e., the right-hand side of (3.3) in the following manner: first, we consider the case where \(f(t) = t^{N}\), a monomial in t of degree \(N \ge 0\). We expand \(\mathscr {P}^{\mathbb {L}}(t, y_{j})\) as \(\sum _{\ell _{j} = 0}^{\infty } \mathscr {P}_{\ell _{j}} (y_{j}) t^{\ell _{j}}\) for \(j = 1, \ldots , n\). Then, we compute

$$\begin{aligned} \dfrac{t^{N}}{\mathscr {P}^{\mathbb {L}} (t) \prod _{j=1}^{n} (t +_{\mathbb {L}} \overline{y}_{j})}= & {} t^{N} \times \dfrac{1}{\mathscr {P}^{\mathbb {L}}(t)} \times \dfrac{1}{ \prod _{j=1}^{n} \dfrac{t - y_{j}}{\mathscr {P}^{\mathbb {L}} (t, y_{j})} } \\= & {} t^{N - n} \times \dfrac{1}{\mathscr {P}^{\mathbb {L}} (t)} \times \displaystyle {\prod _{j=1}^{n}} \mathscr {P}^{\mathbb {L}} (t, y_{j}) \times \prod _{j=1}^{n} \dfrac{1}{1 - y_{j}t^{-1}}\\= & {} t^{N-n} \times \left( \displaystyle {\sum _{k=0}^{\infty }} [\mathbb {C}P^{k}] t^{k} \right) \times \displaystyle {\prod _{j=1}^{n}} \left( \sum _{\ell _{j} = 0}^{\infty } \mathscr {P}_{\ell _{j}} (y_{j}) t^{\ell _{j}} \right) \\&\times \left( \sum _{m=0}^{\infty } h_{m} ({\varvec{y}}_{n}) t^{-m} \right) . \end{aligned}$$

For brevity, we put

$$\begin{aligned} \prod _{j=1}^{n} \left( \sum _{\ell _{j} = 0}^{\infty } \mathscr {P}_{\ell _{j}} (y_{j}) t^{\ell _{j}} \right) = \sum _{\ell = 0}^{\infty } \left( \sum _{ \begin{array}{c} \ell _{1} + \cdots + \ell _{n} = \ell \\ \ell _{1} \ge 0, \ldots , \ell _{n} \ge 0 \end{array} } \prod _{j=1}^{n} \mathscr {P}_{\ell _{j}}(y_{j}) \right) t^{\ell } = \sum _{\ell = 0}^{\infty } \mathscr {P}_{\ell }({\varvec{y}}_{n}) t^{\ell }. \end{aligned}$$

Then, the computation continues as

$$\begin{aligned}&t^{N-n} \times \left( \displaystyle {\sum _{k=0}^{\infty }} [\mathbb {C}P^{k}] t^{k} \right) \times \left\{ \displaystyle {\sum _{r= -\infty }^{\infty }} \left( \sum _{\ell = r}^{\infty } \mathscr {P}_{\ell }({\varvec{y}}_{n}) h_{\ell - r}({\varvec{y}}_{n}) \right) t^{r}\right\} \\&\quad = t^{N - n} \times \left( \displaystyle {\sum _{k=0}^{\infty }} [\mathbb {C}P^{k}] t^{k} \right) \times \left\{ \displaystyle {\sum _{r=0}^{\infty }} \left( \sum _{\ell = r}^{\infty } \mathscr {P}_{\ell } ({\varvec{y}}_{n}) h_{\ell - r} ({\varvec{y}}_{n}) \right) t^{r} + \displaystyle {\sum _{r=1}^{\infty }} \left( \sum _{\ell = 0}^{\infty } \mathscr {P}_{\ell }({\varvec{y}}_{n}) h_{\ell + r}({\varvec{y}}_{n}) \right) t^{-r}\right\} \\&\quad = t^{N - n} \times \left[ \displaystyle {\sum _{s = 0}^{\infty }} \left\{ \sum _{k= 0}^{s} \sum _{\ell = s - k}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell } ({\varvec{y}}_{n}) h_{k + \ell - s} ({\varvec{y}}_{n})\right\} t^{s} \right. \\&\qquad \left. + \displaystyle {\sum _{s = -\infty }^{\infty }} \left\{ \sum _{k = s + 1}^{\infty } \sum _{\ell = 0}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell }({\varvec{y}}_{n}) h_{k +\ell - s}({\varvec{y}}_{n}) \right\} t^{s} \right] \\&\quad = t^{N - n} \times \left[ \displaystyle {\sum _{s = 0}^{\infty }} \left\{ \sum _{k= 0}^{s} \sum _{\ell = s - k}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell } ({\varvec{y}}_{n}) h_{k + \ell - s} ({\varvec{y}}_{n})\right\} t^{s} \right. \\&\qquad + \displaystyle {\sum _{s = 0}^{\infty }} \left\{ \sum _{k = s + 1}^{\infty } \sum _{\ell = 0}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell }({\varvec{y}}_{n}) h_{k +\ell - s} ({\varvec{y}}_{n}) \right\} t^{s}\\&\qquad \left. + \displaystyle {\sum _{s = 1}^{\infty }} \left\{ \sum _{k = 0}^{\infty } \sum _{\ell = 0}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell }({\varvec{y}}_{n}) h_{k +\ell + s} ({\varvec{y}}_{n}) \right\} t^{-s} \right] . \end{aligned}$$

Next, we extract the coefficient of \(t^{-1}\) in the above formal Laurent series. In the case of \(N \ge n\), one sees immediately that the coefficient of \(t^{-1}\) is

$$\begin{aligned} \sum _{k = 0}^{\infty } \sum _{\ell = 0}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell }({\varvec{y}}_{n}) h_{k +\ell + N - n + 1} ({\varvec{y}}_{n}). \end{aligned}$$

In the case of \(0 \le N < n\), the coefficient of \(t^{-1}\) is

$$\begin{aligned}&\displaystyle {\sum _{k= 0}^{n - N - 1}} \sum _{\ell = n - N - 1 - k}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell } ({\varvec{y}}_{n}) h_{k + \ell + N - n + 1} ({\varvec{y}}_{n})\\&+ \sum _{k = n - N }^{\infty } \sum _{\ell = 0}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell }({\varvec{y}}_{n}) h_{k +\ell + N - n + 1} ({\varvec{y}}_{n}) \\&\quad = \displaystyle {\sum _{k= 0}^{\infty }} \sum _{\ell = n - N - 1 - k}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell } ({\varvec{y}}_{n}) h_{k + \ell + N - n + 1} ({\varvec{y}}_{n}). \end{aligned}$$

Summing up the above calculation, we get the following result:

$$\begin{aligned} \underset{t = 0}{\mathrm {Res}'} \dfrac{t^{N}}{\mathscr {P}^{\mathbb {L}} (t) \prod _{j=1}^{n} (t +_{\mathbb {L}} \overline{y}_{j}) } = \left\{ \begin{array}{ll} &{} \displaystyle {\sum _{k = 0}^{\infty }} \sum _{\ell = 0}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell }({\varvec{y}}_{n}) h_{k +\ell + N - n + 1} ({\varvec{y}}_{n}) \quad (N \ge n), \\ &{} \displaystyle {\sum _{k= 0}^{\infty }} \sum _{\ell = n - N - 1 - k}^{\infty } [\mathbb {C}P^{k}] \mathscr {P}_{\ell } ({\varvec{y}}_{n}) h_{k + \ell + N - n + 1} ({\varvec{y}}_{n}) \quad (0 \le N < n). \end{array}\right. \end{aligned}$$
(5.1)

For a general polynomial of the form \(f(t) = \sum _{N=0}^{M} a_{N} t^{N} \in MU^{*}(X)[t]\), one has \(\varpi _{1 *} (f(y_{1})) = \varpi _{1 *} \left( \sum _{N=0}^{M} a_{N} y_{1}^{N} \right) = \sum _{N=0}^{M} a_{N} \varpi _{1 *} (y_{1}^{N})\) since the Gysin map \(\varpi _{1 *}\) is an \(MU^{*}(X)\)-homomorphism. From this, we can calculate \(\varpi _{1 *}(f(y_{1}))\).

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Nakagawa, M., Naruse, H. Darondeau–Pragacz formulas in complex cobordism. Math. Ann. 381, 335–361 (2021). https://doi.org/10.1007/s00208-021-02196-5

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