# Hirzebruch *L*-polynomials and multiple zeta values

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## Abstract

We express the coefficients of the Hirzebruch *L*-polynomials in terms of certain alternating multiple zeta values. In particular, we show that every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign. Similar results hold for the polynomials associated to the \(\hat{A}\)-genus.

## Mathematics Subject Classification

Primary 55R40 11M32 Secondary 57R20## 1 Introduction

*L*-polynomials are certain polynomials with rational coefficients,

*M*, see [3, Theorem 8.2.2] or [5, Theorem 19.4]. The

*k*th polynomial has the form

*k*, i.e., sequences of integers \(j_1\ge \cdots \ge j_r \ge 1\) such that \(j_1+\cdots + j_r =k\). The purpose of this note is to establish certain properties of the coefficients \(h_{j_1,\ldots ,j_r}\).

*n*is even”. Define the symmetrization of this series by

### Theorem 1

*L*-polynomials are given by

It is well-known that \(h_k\) is positive for all *k*. In [8, Appendix A], it is argued that \(h_{i,j}\) is always negative and that \(h_{i,j,k}\) is always positive (following an argument attributed to Galatius in the case of \(h_{i,j}\)), and it is asked whether it has been proved in general that \((-1)^{r-1}h_{j_1,\ldots ,j_r}\) is positive. We have not been able to locate such a result in the literature, but we can prove it using our formula. It follows from the following result.

### Theorem 2

### Corollary 3

The coefficient \(h_{j_1,\ldots ,j_r}\) in the Hirzebruch *L*-polynomial \(\mathsf L_k\) is non-zero for every partition \((j_1,\ldots ,j_r)\) of *k*. It is negative if *r* is even and positive if *r* is odd.

*k*. Consider the series

### Theorem 4

*r*is odd and positive if

*r*is even.

## 2 Proofs

The first step in our proof is to establish a formula that expresses the coefficient \(h_{j_1,\ldots ,j_r}\) as a linear combination of products \(h_{k_1}\cdots h_{k_\ell }\). This generalizes the formulas for \(h_{i,j}\) and \(h_{i,j,k}\) found in [8]. In the appendix of [2], recursive formulas for computing \(h_{j_1,\ldots ,j_r}\) in terms of products \(h_{k_1}\cdots h_{k_\ell }\) are given. Here we give an explicit closed formula. The result holds for arbitrary multiplicative sequences of polynomials (see [3, §1]).

### Theorem 5

*i*, the sum is over all partitions \(\mathcal {P} = \{P_1,\ldots ,P_\ell \}\) of the set \(\{1,2,\ldots ,r\}\),

### Proof

*monomial symmetric function*in \(\beta _1',\ldots ,\beta _m'\) (see [3, Lemma 1.4.1]).

Note that \(\lambda _k\) equals the power sum \(\sum _i (\beta _i')^k\). The product \(\lambda _{k_1}\cdots \lambda _{k_\ell }\) is then the *power sum symmetric function* evaluated at \(\beta _i'\), and the claim follows from a general formula that expresses the monomial symmetric functions in terms of power sum symmetric functions, see Theorem 8 below. \(\square \)

*L*-polynomials is

*k*,

*n*is even”. Then symmetrize, and define

### Theorem 6

### Proof

This will follow by specialization of Theorem 10 below. \(\square \)

### Proof of Theorem 2

*r*. The first equality then shows that \(T(s_1,\ldots ,s_r)\) is negative. \(\square \)

*k*. By using the Cauchy formula (see [3, p.11]) one can calculate the coefficient \(a_k\) of \(\mathsf {p}_k\) in \({\hat{\mathsf {A}}}_k\). The result is

*k*. Theorem 5 then yields

*k*. However, more can be said; the sum in the right hand side now not only resembles but is

*equal*to the right hand side of another formula of Hoffman [4, Theorem 2.1]. In our notation this formula says that

## 3 Combinatorics of infinite sums

The proofs of Theorems 5 and 6, as well as of Hoffman’s formula, share the same combinatorial underpinnings; this is the topic of the present section.

*partition*of a set

*S*is a set of non-empty disjoint subsets,

*length*of \(\pi \). The set of partitions \(\Pi _S\) is partially ordered by refinement, \(\pi = \{\pi _1\ldots ,\pi _r\} \le \rho = \{\rho _1,\ldots ,\rho _\ell \}\) if and only if there is a partition \(\mathcal {P}= \{P_1,\ldots ,P_\ell \}\) of the set \(\{1,2,\ldots ,r\}\) such that

*n*. For a subset \(T\subseteq S\), write

*S*, consider the formal power series

### Theorem 7

### Proof

Take \(S= \{1,2,\ldots ,r\}\) and substitute \(a_n\) by \(\frac{1}{n^{s_a}}\) for \(a\in S\) in (8). \(\square \)

*power sum symmetric function*\(p_I\) is the formal power series in indeterminates \(x_1,x_2,\ldots \) defined by \(p_I = p_{i_1}\cdots p_{i_r}\), where

*monomial symmetric function*\(m_I\) is defined as the sum of all pairwise distinct monomials of the form \(x_{\sigma _1}^{i_1}\cdots x_{\sigma _r}^{i_r}\).

### Theorem 8

*k*,

*j*, and \(J = (j_1,\ldots ,j_\ell )\) is given by

### Proof

Let *S* be any set with *k* elements. Perform the substitution \(a_n = x_n\) for each \(a\in S\) in the equality (8) and note that this takes \(p_\pi \) to \(p_I\) and \(m_\pi \) to \(\alpha _1!\cdots \alpha _k!\, m_I\), where \(I = \big (|\pi _1|,\ldots ,|\pi _r|\big )\) is the integer partition underlying the set partition \(\pi \) (assuming, as we may, \(|\pi _1|\ge \cdots \ge |\pi _r|\)). \(\square \)

*ordered*partition \(\widetilde{\pi } = (\pi _1,\ldots ,\pi _r)\) we define

*r*! ordered partitions \(\widetilde{\pi }\) whose underlying unordered partition is \(\pi \).

### Lemma 9

### Proof

*S*. The number of partitions of length

*k*in \(\Pi _S\) is equal to the Stirling number of the second kind

*S*(

*n*,

*k*), see e.g. [6, Example 3.10.4]. Thus,

### Theorem 10

### Proof

*e*is an assignment of a parity \(e_i \in \{0,1\}\) to each \(\nu _i\). For example, if \(\nu = \{\{a,b\},\{c\}\}\) and

*e*assigns 1 to \(\{a,b\}\) and 0 to \(\{c\}\), then

*e*assigns an even value to \(\nu _i\) whenever \(\nu _i\) consists of more than one \(\rho \)-block. This can be reformulated as saying that \(\nu \ge \rho \ge e(\nu )\), where \(e(\nu )\le \nu \) is the partition that keeps \(\nu _i\) intact if \(e_i\) is odd and splits \(\nu _i\) completely if \(e_i\) is even. Or more precisely, \(e(\nu )\) is the smallest element below \(\nu \) that contains \(\nu _i\) whenever \(e_i\) is odd. Since we symmetrize, there will be repetitions; for \(\nu \ge \rho \ge e(\nu )\), the term involving \(m_{\nu ,e}\) will be repeated \(b_{\rho ,\nu } = b_1! \cdots b_m!\) times in \(T_\rho ^\Sigma \), where \(b_i\) is the number of \(\rho \)-blocks in \(\nu _i\). Thus, the coefficient of \(m_{\nu ,e}\) in \((-1)^{\ell (\rho )} T_\rho ^\Sigma \) is \({\text {sgn}}(\rho ,\nu ,e)b_{\rho ,\nu }\). It follows that the coefficient of \(m_{\nu ,e}\) in \(\sum _{\rho \ge \pi } (-1)^{\ell (\rho )} T_\rho ^\Sigma \) is

To prove Theorem 6, take \(S=\{1,2,\ldots ,r\}\) and substitute \(a_n\) by \(\frac{1}{n^{s_a}}\) in (10).

## Notes

### Acknowledgements

We thank Don Zagier and Matthias Kreck for valuable comments. The impetus for this work was a question from Oscar Randal-Williams to the first author about certain points in [1]. The first author was supported by the Swedish Research Council through Grant No. 2015-03991.

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