# Systematic analysis of the nuclear absorption effect on the cross section of
the knockout reaction

Sang-In Shim<EMAIL_ADDRESS>Center for Extreme Nuclear Matters (CENuM),
Korea University, Seoul 02841, Republic of Korea  Research Center for Nuclear
Physics (RCNP), Osaka University, Ibaraki, Osaka, 567-0047, Japan  Kazuki
Yoshida  Advanced Science Research Center, Japan Atomic Energy Agency, Tokai,
Ibaraki 319-1195, Japan  Kazuyuki Ogata  Department of Physics, Kyushu
University, Fukuoka 819-0395, Japan  Research Center for Nuclear Physics
(RCNP), Osaka University, Ibaraki, Osaka, 567-0047, Japan

###### Abstract

Recent studies on nucleon and alpha knockout reactions have shown that the
distorted-wave impulse approximation (DWIA) is a simple and accurate method to
describe these reactions. As it has been argued for decades, the nuclear
absorption is one of the most important ingredients of the DWIA calculation.
In this work, we systematically investigate the absorption effects on the
cross sections of the nucleon and alpha knockout reactions. To do this, we
calculate the ratio of the cross sections of the DWIA and plane-wave impulse
approximation (PWIA) and examine its dependence on the mass number and single-
particle orbital of the knocked-out particles. We will discuss the specific
characteristics of the absorption effect for each reaction.

##  I Introduction

For more than half a century, the proton-induced nucleon knockout reaction,
($p,p$N), has been a powerful spectroscopic tool for studying the single-
particle (s.p.) picture of nuclei Jacob and Maris (1966, 1973); Kitching  _et
al._ (1985); see also reviews Wakasa  _et al._ (2017); Aumann  _et al._
(2021). In recent years, the s.p. nature and the shell evolution Otsuka  _et
al._ (2001, 2005); Otsuka (2013); Otsuka and Tsunoda (2016) in neutron-rich
nuclei have been intensively studied at RIKEN RIBF Santamaria  _et al._
(2015); Paul  _et al._ (2017); Flavigny  _et al._ (2017); Chen  _et al._
(2017); Lettmann  _et al._ (2017); Shand  _et al._ (2017); Olivier  _et al._
(2017); Cortés  _et al._ (2018); Liu  _et al._ (2018, 2019); Paul  _et al._
(2019); Taniuchi  _et al._ (2019); Elekes  _et al._ (2019); Chen  _et al._
(2019); Cortés  _et al._ (2020, 2020); Sun  _et al._ (2020); Lokotko  _et al._
(2020); Lizarazo  _et al._ (2020); Frotscher  _et al._ (2020); Juhász  _et
al._ (2021a, b); Browne  _et al._ (2021); Linh  _et al._ (2021); Koiwai  _et
al._ (2022); Gerst  _et al._ (2022); Enciu  _et al._ (2022); Elekes  _et al._
(2022). It should also be mentioned that regarding the quenching of the
spectroscopic strength Gade  _et al._ (2008); Tostevin and Gade (2014), the
nucleon knockout reaction of a wide range of carbon, nitrogen, and oxygen
isotopes has been measured at the R3B-LAND setup at GSI Panin  _et al._
(2016); Atar  _et al._ (2018); Díaz Fernández  _et al._ (2018); Holl  _et al._
(2019) and RIKEN Kawase  _et al._ (2018). In Ref. Phuc  _et al._ (2019), these
data were reanalyzed using the distorted-wave impulse approximation (DWIA)
framework Chant and Roos (1977, 1983), and various corrections and
uncertainties within the standard DWIA formalism on the ($p,p$N) cross
sections were systematically investigated.

In the determination of the s.p. wave function using the knockout reaction
cross section, the absorption effect in the distorted waves of the scattering
particles plays a crucial role. Since the absorption makes the reaction
peripheral, its effect on the reaction observables depends on the s.p.
property of a struck nucleon, e.g., its orbital angular momentum $l$, radial
quantum number $n$, nucleon separation energy, and its spatial distribution.
Similarly to ($p,p$N), the proton induced $\alpha$ knockout reaction,
($p,p\alpha$), is a suitable probe for the $\alpha$ clustering in nuclei
Yoshida  _et al._ (2019); Taniguchi  _et al._ (2021). It is shown that the
($p,p\alpha$) reaction is peripheral even in the case of light nuclei,
20Ne($p,p\alpha$)16O Yoshida  _et al._ (2019), and there is almost no
contribution from the internal region in medium and heavy mass cases Yoshida
_et al._ (2016, 2018); Taniguchi  _et al._ (2021); Yoshida  _et al._ (2022).
This is because the mean-free-path of $\alpha$ in a nucleus is short and the
absorption effect is much stronger than that of a nucleon.

In Sec. 5.5. and Fig. 31 of Ref. Wakasa  _et al._ (2017), the absorption
effect has been discussed using the ratio of the DWIA and PWIA (plane-wave
impulse approximation) s.p. cross sections. However, the previous work
included only a small number of cases. Therefore, a systematic analysis of the
absorption effect and its s.p. orbital dependence is required. For this
purpose, in this paper, we investigate the ratio of the DWIA and PWIA cross
sections (the DW-PW ratio) of the nucleon and the $\alpha$ knockout reactions
in more diverse atomic nuclei, particularly focusing on the s.p. and cluster
orbital dependence. It can be shown that the PWIA cross section roughly
corresponds to the total elastic cross section of the elementary process Ogata
_et al._ (2015); Gómez-Ramos . Systematic knockout reaction experiments are
planned for the future, and the DW-PW ratios in this study will be helpful in
estimating the cross sections of the knockout reaction from those of the
elementary process prior to the actual experiments.

The content of this paper is as follows. In section II, we briefly explain the
framework of DWIA, PWIA, and the cross sections derived from those methods.
The numerical inputs for the ($p$,$p$N) and ($p$,$p\alpha$) reactions and
information about the target nuclei and the s.p. orbitals are also given
there. In section III, we present the numerical results of the DW-PW ratio for
the ($p$,$p$N) and ($p$,$p\alpha$) reactions. We discuss the target, s.p.
orbital, and incident energy dependences of the DW-PW ratio. The final section
is devoted to summary and conclusion.

##  II DWIA framework

###  II.1 Knockout cross section

The factorized form of the DWIA framework is used for the present
calculations. Since we follow the DWIA calculation used in Refs. Wakasa  _et
al._ (2017); Yoshida and Tanaka (2022), only essential parts are described
below. The triple differential cross section (TDX) is given by

$\displaystyle\frac{d^{3}\sigma}{dE_{1}^{\mathrm{A}}d\Omega_{1}^{\mathrm{A}}d\Omega_{\mathrm{C}}^{\mathrm{A}}}$
$\displaystyle=F_{\mathrm{kin}}^{\mathrm{A}}\frac{E_{1}E_{\mathrm{C}}E_{\mathrm{B}}}{E_{1}^{\mathrm{A}}E_{\mathrm{C}}^{\mathrm{A}}E_{\mathrm{B}}^{\mathrm{A}}}\frac{(2\pi)^{4}}{\hbar
v_{\alpha}}\frac{1}{2l+1}\frac{(2\pi\hbar^{2})^{2}}{\mu_{p\mathrm{C}}^{2}}\frac{d\sigma_{p\mathrm{C}}}{d\Omega_{p\mathrm{C}}}\left|\bar{T}\right|^{2}.$
(1)

The labels $0,1$, and $\mathrm{B}$ denote the incoming proton, the emitted
proton, and the residual nucleus, respectively, and $\mathrm{C}$ is the struck
particle ($\mathrm{C}=$ N or $\alpha$). $E_{i}$ and $\Omega_{i}$ are the total
energy and the solid angle of the emission direction of the outgoing particle
$i=1,\mathrm{B}$ or $\mathrm{C}$. Quantities with a superscript $\mathrm{A}$
are evaluated in the nucleus rest frame (A frame), while the others are
evaluated in the center-of-mass frame of the reaction. $v_{\alpha}$, $l$, and
$\mu_{p\mathrm{C}}$ are the relative velocity of the initial proton and the
target, the orbital angular momentum of the $\mathrm{B}$-$\mathrm{C}$ bound
state, and the reduced mass of the proton and $\mathrm{C}$, respectively. The
phase volume factor for this expression is denoted by
$F_{\mathrm{kin}}^{\mathrm{A}}$; see Ref. Yoshida and Tanaka (2022) for the
explicit expression. $d\sigma_{p\mathrm{C}}/d\Omega_{p\mathrm{C}}$ is the
differential cross section of the $p$-$\mathrm{C}$ elastic scattering; the
energy and the scattering angle of the $p$-$\mathrm{C}$ binary scattering are
determined from the ($p,p\mathrm{C}$) kinematics.

The reduced transition matrix $\bar{T}$ is given by

$\displaystyle\bar{T}$
$\displaystyle=\int\chi_{1}^{*(-)}(\bm{R})\chi_{2}^{*(-)}(\bm{R})\varphi_{\mathrm{C}}(\bm{R})\chi_{0}^{(+)}(\bm{R})e^{-\bm{K}_{0}\cdot\bm{R}A_{\mathrm{C}}/A}d\bm{R},$
(2)

where $\chi_{0}$, $\chi_{1}$, and $\chi_{2}$ are the distorted waves between
$p$-A, $p$-B, and C-B, respectively. The superscripts $(+)$ and $(-)$ denote
the outgoing and incoming boundary conditions, respectively.
$\varphi_{\mathrm{C}}$ is either the nucleon s.p. wave function ($\mathrm{C}=$
N) or the relative wave function between the $\alpha$ cluster and the residual
nucleus B ($\mathrm{C}=\alpha$). $\bm{K}_{0}$ is the momentum (wave number) of
the incident proton. The mass numbers of particle $\mathrm{C}$ and the target
nucleus A are given by $A_{\mathrm{C}}$ and $A$, respectively. The total
knockout cross sections discussed below are obtained by integrating Eq. (1)
over all possible kinematics. In PWIA calculations, all the distorted waves in
Eq. (2) are replaced with the corresponding plane waves.

The DW-PW ratio is defined by

$\displaystyle\frac{\sigma^{\mathrm{DW}}}{\sigma^{\mathrm{PW}}}$
$\displaystyle=\frac{\displaystyle\int\left(\displaystyle\frac{d^{3}\sigma}{dE^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{\mathrm{C}}}\right)dE^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{\mathrm{C}}}{\displaystyle\int\left(\displaystyle\frac{d^{3}\sigma^{\mathrm{PW}}}{dE^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{\mathrm{C}}}\right)dE^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{\mathrm{C}}},$
(3)

where
$d^{3}\sigma^{\mathrm{PW}}/dE^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{1}d\Omega^{\mathrm{A}}_{\mathrm{C}}$
is the same as the TDX given by Eq. (1) but calculated with PWIA.

###  II.2 Numerical input

For ($p,p$N) calculations, proton beams at 50 MeV and 200 MeV are considered.
The nucleon s.p. wave function $\varphi_{N}$ is obtained using the Woods-Saxon
shaped potential proposed by Bohr and Mottelson Bohr and Mottelson (1969). The
depth of the potential is adjusted to reproduce the nucleon separation energy.
The global optical potential for a wide range of the proton nucleus scattering
by Koning and Delaroche Koning and Delaroche (2003) is employed to calculate
the nucleon-nucleus distorted waves. The $p$-N differential cross section is
calculated using the Franey-Love effective NN interaction Franey and Love
(1985). For the target nuclei, we consider 12C, 40Ca, 58Ni, 76Se, 90Zr, 108Cd,
120Sn, 144Er, 186Os, 196Pt, and 208Pb. We calculate the DW-PW ratio of Eq. (3)
for the 0p, 0d, 0f, 0g, 1p, 2s, and 2p nucleon s.p. orbitals, corresponding to
($n$,$l$) = (0,1), (0,2), (0,3), (0,4), (1,1), (2,0), and (2,1), respectively.

For the ($p,p\alpha$) calculations, following the former experiment Carey  _et
al._ (1984), ($p,p\alpha$) reactions at 101.5 MeV on the target nuclei 12C,
16O, 20Ne, 24Mg, 28Si, 32S, 40Ca, 48Ti, 54Fe, 66Zn, 90Zr, 120Sn, and 208Pb are
considered. As mentioned in the previous subsection, the framework for the
($p,p\alpha$) reaction is basically the same as the ($p,p$N) reaction, except
that the s.p. wave function $\varphi_{\mathrm{N}}$ is replaced with
$\varphi_{\mathrm{\alpha}}$. $\varphi_{\alpha}$ is obtained as a bound-state
wave function of the Woods-Saxon potential. Its range parameter
$r_{0}=1.25B^{1/3}$ fm and the diffuseness parameter $a_{0}=0.76$ fm are taken
from Ref. Fukui  _et al._ (2016). Here, $B=A-4$ is the mass number of the core
(residual) nucleus. The optical potential parameters proposed by Avrigeanu and
his collaborators Avrigeanu  _et al._ (1994) are used for the distorted wave
of the emitted $\alpha$. The $p$-$\alpha$ differential cross section in free
space is calculated by the folding model Toyokawa  _et al._ (2013) using the
Melbourne $g$-matrix interaction Amos  _et al._ (2000) and a phenomenological
$\alpha$ density. For $\varphi_{\alpha}$, $l=0$ is assumed, and the radial
quantum number (the number of nodes) $n$ is determined such that the
$\mathrm{B}+\alpha$ bound state has the same total harmonic oscillator quantum
numbers of nucleus A in the ground state. The specific values of $n$ are
listed in Table 1.

Table 1: The target nucleus and the radial quantum number $n$ of $\varphi_{\alpha}$ for the ($p,p\alpha$) knockout reaction. Nucleus |  12C |  16O |  20Ne |  24Mg |  28Si |  32S |  40Ca |  48Ti |  54Fe |  66Zn |  90Zr |  120Sn |  208Pb
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$n$ | 2 | 4 | 6 | 8 | 11

##  III Results and discussion

###  III.1 Absorption effect and DW-PW ratio

The DW-PW ratio of the ($p,p$N) cross sections at 50 and 200 MeV are shown in
Figs. 1 and 2, respectively. It is found that the heavier the atomic nucleus,
the smaller the DW-PW ratio, which is consistent with the result of the
aforementioned Fig. 31 of Ref. Wakasa  _et al._ (2017). It can be also seen
that the DW-PW ratio becomes larger as $n$ and $l$ increase, and particularly,
it is more sensitive to changes in $n$. The DW-PW ratio of the ($p,p\alpha$)
cross sections at 101.5 MeV is shown in Fig. 3. Its dependence on the mass
number $A$ and $n$ is similar to that for ($p,p$N). These features of the DW-
PW ratio can be understood as follows.

For given $n$ and $l$, the absorption effect becomes larger as $A$ increases.
This is because the nonelastic processes for the scattering particles, i.e.,
particles 0, 1, and C, become more important for heavier target nuclei,
reflecting the increase in the degree of freedom of the reaction systems. This
general property of the DW-PW ratio can be seen in Figs. 1, 2, and 3. The only
exception is the 2p orbital in the ($p$,$pn$) knockout reaction at 50 MeV; we
will return to this point below. The dependence of the DW-PW ratio for
($p,p$N) on the incident energy $E_{\rm in}$ can be understood in the same way
as above. Namely, particles 0, 1, and C can easily penetrate the nucleus at
higher $E_{\rm in}$. The absolute values of the DW-PW ratio in Fig. 2 are
noticeably larger than in Fig. 1.

As for the s.p. orbital dependence, the increase in the DW-PW ratio for higher
$n$ is confirmed in the ($p$,$p$N) reactions at 50 MeV and 200 MeV and
($p$,$p\alpha$) at 101.5 MeV. This can be understood from the form of
$\varphi_{N}$ and $\varphi_{\alpha}$. Their probability densities increase
near the surface of the nucleus as $n$ increases. For similar reasons, the DW-
PW ratio becomes larger when $l$ increases, but the change is not as
significant as that due to an increase in $n$. For $n=0$, the $l$-dependence
of the DW-PW ratio at 50 MeV is found to be less noticeable compared to at 200
MeV.

For intuitive understanding of these features of the DW-PW ratio, let us
introduce two radii: the absorption radius $r_{\rm abs}$ and the radius
$r_{\rm C}$ of the nucleon or alpha particle to be knocked out. Here, we
assume $r_{\rm C}$ is same as the root mean square radius of $\varphi_{N}$ and
$\varphi_{\alpha}$. Then, one may use the following geometrical model for the
DW-PW ratio:

$\displaystyle\frac{\sigma^{\mathrm{DW}}}{\sigma^{\mathrm{PW}}}$
$\displaystyle\sim\frac{r^{3}_{\rm C}-r^{3}_{\rm abs}}{r^{3}_{\rm
C}}=1-\frac{r^{3}_{\rm abs}}{r^{3}_{\rm C}}.$ (4)

When we consider specific $n$ and $l$, $r_{\rm C}$ can be assumed to be a
constant because its $A$-dependence is usually weak within a certain rage of
$A$. As $A$ becomes larger or $E_{\rm in}$ becomes lower, $r_{\rm abs}$
increases, resulting in a smaller DW-PW ratio. On the other hand, for a
specific $A$ and $E_{\rm in}$, $r_{\rm abs}$ can be regarded as a constant.
Because $r_{\rm C}$ becomes larger as $n$ or $l$ increases, one obtains a
larger DW-PW ratio. Now, let us consider the exceptional behavior of the DW-PW
ratio for 2p neutron knockout at 50 MeV. In Table 2, we list $r_{\rm C}$ for
2p neutron considered in Fig. 1 as well as for 2s neutron and 2s proton in the
same mass region.

Table 2: The knocked-out nucleon radius $r_{\rm C}$ for the $A>150$ nuclei. Nucleus | $r_{\mathrm{C}}$
---|---
2p neutron | 2s neutron | 2s proton
166Er | $\cdot$ | 5.77 fm | 5.25 fm
186Os | 6.16 fm | 5.86 fm | 5.35 fm
196Pt | 6.25 fm | $\cdot$ | $\cdot$
208Pb | 6.37 fm | $\cdot$ | 5.42 fm

As seen, while the $A$-dependence of $r_{\rm C}$ for the 2s proton and 2s
neutron is weak, that for the 2p neutron is exceptionally strong. This makes
the behavior of the DW-PW ratio rather nontrivial, meaning that we need to
consider the $A$-dependence of not only $r_{\rm abs}$ but also $r_{\rm C}$.
The results shown in Figs. 1 and 2 suggest that the $A$-dependence of $r_{\rm
C}$ is more (less) important than of $r_{\rm abs}$ for the 2p neutron DW-PW
ratio at 50 MeV (200 MeV).

Figure 1: DW-PW ratio of ($p$,$p$N) at 50 MeV. Figure 2: DW-PW ratio of
($p$,$p$N) at 200 MeV. Figure 3: DW-PW ratio of ($p$,$p\alpha$) at 101.5 MeV.

###  III.2 The separation of the ($p$,$2p$) and ($p$,$pn$) ratios

The difference between the DW-PW ratios of ($p$,2$p$) and ($p$,$pn$) appears
in most s.p. orbitals at 50 MeV and in the 2s orbital at 200 MeV. There are
two main causes of this separation. One is the Coulomb potential for scattered
nucleons, which makes it more difficult for protons to penetrate the nucleus
at low energies, in particular, resulting in the reduction of the DW-PW ratio
for ($p$,2$p$). At 50 MeV, this is the case with most orbitals, whereas, at
200 MeV, this effect is negligibly small. The other cause of the separation of
($p$,2$p$) and ($p$,$pn$) is the s.p. wave function of the bound nucleons. In
Fig. 4, some specific s.p. wave functions with $n$ = 0, 1, and 2 are shown. As
the mass number increases, the difference in the s.p. wave functions of
protons and neutrons develops because the Coulomb barrier suppresses the
proton s.p. wave function at the nuclear surface. This causes the separation
of the ($p$,2$p$) and ($p$,$pn$) for the 2s orbitals at 200 MeV.

The importance of these causes depends on the incident energy and $n$. More
specifically, the Coulomb potential provides the separation of $n$ = 0
orbitals at 50 MeV, and the difference in the bound-state wave functions gives
the separation of the $n$ = 2 orbitals at 200 MeV. For the separation of $n$ =
1 and 2 orbitals at 50 MeV, these effects are combined.

Figure 4: Some selected s.p. wave functions $r^{2}R(r)$ of $n$ = 0, 1, 2.
$R(r)$ is the radial part of the s.p. wave functions, and $r^{2}$ originates
from $d\bm{R}$ in Eq. (2).

##  IV Summary and conclusions

We conducted a systematic investigation of the absorption effect depending on
the mass number and the orbitals of the bound proton, neutron, and $\alpha$ by
calculating the ratio of the scattering cross sections of the DWIA and PWIA,
i.e., the DW-PW ratio, for the ($p$,$p$N) and ($p$,$p\alpha$) reactions. Since
the PWIA cross section $\sigma^{\rm PW}$ is similar to the total elastic cross
section of the elementary process, the DW-PW ratio tells us how much the
knockout cross section is reduced from the cross section of the elementary
process and will be helpful in estimating the experimental knockout cross
section. For the ($p$,$p$N) reaction, the target atomic nuclei 12C, 40Ca,
58Ni, 76Se, 90Zr, 108Cd, 120Sn, 144Er, 186Os, 196Pt, 208Pb, the s.p. orbitals
of the bound nucleons ($n$,$l$) = (0,1), (0,2), (0,3), (0,4), (1,1), (2,0),
(2,1), and the incident proton of 50 MeV and 200 MeV are considered. For the
($p$,$p\alpha$) reaction, the target atomic nucleus 12C, 16O, 20Ne, 24Mg,
28Si, 32S, 40Ca, 48Ti, 54Fe, 66Zn, 90Zr, 120Sn, 208Pb, and the incident proton
of 101.5 MeV are considered. In addition, the orbitals of the bound $\alpha$
with $l=0$ and $n=2$, 4, 6, and 11 are assumed. The mass number and s.p.
orbital dependences of the absorption effect of the atomic nucleus were
discussed by using the DW-PW ratio. As has been confirmed in the former review
Wakasa  _et al._ (2017), in general, the heavier nuclei with higher mass
numbers the smaller the DW-PW ratio in both ($p$,$p$N) and ($p$,$p\alpha$)
reactions. This is because it becomes difficult for nucleons to penetrate the
atomic nucleus, and the absorption effect of the atomic nucleus becomes
stronger. Regarding the dependence on the s.p. orbitals, the DW-PW ratio
increases with increasing $n$ in both ($p$,$p$N) and ($p$,$p\alpha$)
reactions, because the probability density of the s.p. state increases near
the surface of the nucleus as $n$ increases, making it easier to scatter
compared to penetrating the core of the nucleus. For similar reasons, the DW-
PW ratio becomes larger with increasing $l$ in the ($p$,$p$N) reaction, but
this change is less significant than its dependence on $n$ and is difficult to
notice at around 50 MeV. Finally, the difference between the DW-PW ratio of
($p$,2$p$) and ($p$,$pn$) reactions was discussed. At 50 MeV, clear
differences in the DW-PW ratio between ($p$,2$p$) and ($p$,$pn$) reactions are
observed in most s.p. orbitals. This separation is attributed to differences
in the Coulomb potential for the scattered particle, that of the s.p. wave
function, and the combination of these two.

Since the DW-PW ratio investigated in this study is for the s.p. total cross
section, the contributions of each part of the atomic nucleus considering the
angle and depth to the DW-PW ratio of knockout reactions were not fully
understood. Further research on the DW-PW ratio for each part of the nucleus
is needed for a better understanding of the absorption effect of the nucleus,
such as the reversal of the $n=0$ and 1 orbitals at low incident energy Wakasa
_et al._ (2017), and the separations in the 0p and 0d orbitals at the incident
energy of 200 MeV in the present work.

## Acknowledgments

The authors are grateful to Y. Chazono (RIKEN) for valuable and helpful
discussions. The work of S.-I. Shim was supported by the National Research
Foundation of Korea (NRF) grants funded by the Korean government (MSIT)
(2018R1A5A1025563, 2022R1A2C1003964, and 2022K2A9A1A0609176). This work has
been supported in part by Grants-in-Aid of the Japan Society for the Promotion
of Science (Grants No. JP20K14475, No. JP21H00125, and No. JP21H04975).

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# Inverting the wedge map and Gauss composition

Kok Seng Chua<EMAIL_ADDRESS>
###### Abstract.

Let $1\leq k\leq n,$ and let $v_{1},\ldots,v_{k}$ be integral vectors in
${\mathbb{Z}}^{n}$. We consider the wedge map
$\alpha_{n,k}:({\mathbb{Z}}^{n})^{k}/SL_{k}({\mathbb{Z}})\rightarrow\wedge^{k}({\mathbb{Z}}^{n})$,
$(v_{1},\ldots,v_{k})\rightarrow v_{1}\wedge\cdots\wedge v_{k}$. In his
Disquisitiones, Gauss proved that $\alpha_{n,2}$ is injective when restricted
to a primitive system of vectors when defining his composition law for binary
quadratic forms. He also gave an algorithm for inverting $\alpha_{3,2}$ in a
different context on the representation of integers by ternary quadratic
forms. We give here an explicit algorithm for inverting $\alpha_{n,2}$, and
observe via Bhargava’s composition law for
${\mathbb{Z}}^{2}\otimes{\mathbb{Z}}^{2}\otimes{\mathbb{Z}}^{2}$ cube that
inverting $\alpha_{4,2}$ is the main algorithmic step in Gauss’s composition
law for binary quadratic forms. This places Gauss’s composition as a special
case of the geometric problem of inverting a wedge map which may be of
independent interests. We also show that a given symmetric positive definite
matrix $A$ induces a natural metric on the integral Grassmannian
$G_{n,k}({\mathbb{Z}})$ so that the map $X\rightarrow X^{T}AX$ becomes norm
preserving.

###### Key words and phrases:

Gauss’s composition, wedge product, quadratic forms, Bhargava’s cube, Plücker
coordinates, integral Grassmannian

###### 2000 Mathematics Subject Classification:

Primary : 11-02, Secondary : 11E16,11Y16

##  1\. Introduction

Let $1\leq k\leq n$ and $v_{1},\ldots,v_{k}\in{\mathbb{Z}}^{n}$, consider the
$k$-wedge product map

$X=(v_{1},\cdots,v_{k})\rightarrow Y=v_{1}\wedge\cdots\wedge
v_{k}\in\wedge^{k}{\mathbb{Z}}^{n}\cong{\mathbb{Z}}^{n_{k}},\;n_{k}=\begin{pmatrix}n\cr
k\end{pmatrix}$

where for the index $I=i_{1}\cdots i_{k},\;\;1\leq
i_{1}<i_{2}<\cdots<i_{k}\leq n,$ the $I$th Plücker coordinate of $Y$ is given
by

(1.1)
$Y_{I}=\det\begin{pmatrix}(v_{1})_{i_{1}}&\cdots(v_{k})_{i_{1}}\cr...&...\cr(v_{1})_{i_{k}}&\cdots(v_{k})_{i_{k}}\end{pmatrix}.$

Since for any $k\times k$ matrix $A=(a_{ij})$, we have

(1.2) $(\sum_{j}a_{1j}v_{j})\wedge\cdots\wedge(\sum_{j}a_{kj}v_{j})=(\det
A)\;\;v_{1}\wedge\cdots\wedge v_{k},$

the wedge map descends to a map of the orbits of
$({\mathbb{Z}}^{n})^{k}\cong{\mathbb{Z}}^{nk}$ (identifying $X$ with a
$n\times k$ integral matrix) under right multiplication by
$SL_{k}({\mathbb{Z}})$, and writing $G_{n,k}({\mathbb{Z}}):=Im(\alpha_{n,k})$,

(1.3) $\alpha_{n,k}:({\mathbb{Z}}^{n})^{k}/SL_{k}({\mathbb{Z}})\rightarrow
G_{n,k}({\mathbb{Z}})\subset\wedge^{k}{\mathbb{Z}}^{n}.$

In general $\alpha_{n,k}$ is not one-one since for example, we have

$(1,2,2)^{T}\wedge(1,0,4)^{T}=(0,1,-1)^{T}\wedge(2,0,8)^{T}=(8,2,-2)^{T},$

and the two pairs of bi-vectors are not $SL_{2}$ equivalent, but note that the
vector $(8,2,-2)^{T}$ is not primitive. In general we will say that a set of
$k$ vectors $v_{1},\cdots,v_{k}$ in ${\mathbb{Z}}^{n}$ forms a primitive
system if the gcd of the $n\choose k$ components of $v_{1}\wedge\cdots\wedge
v_{k}$ is one.

In his Disquisitiones ([6] Art. 234), where he defined his composition law for
binary quadratic forms, Gauss proved that the restriction of $\alpha_{n,k}$ to
primitive systems of vectors, which we denote by $\alpha_{n,k}^{*}$ is
injective in the case $k=2$. This actually holds for all $k$ and we prove the
case $k=3$ in Lemma 3.4 below which constructs an explicit $SL_{3}$
transformation relating any two pre-images in $({\mathbb{Z}}^{n})^{3}$. The
construction generalizes obviously for arbitrary $k$. Note that geometrically
this just says that the ”Plücker embedding” is indeed an embedding over Spec
${\mathbb{Z}}$.

In his work on the representation of integers by ternary quadratic forms ([6]
Art.279), Gauss gave an algorithm for inverting $\alpha_{3,2}$. This is used
in the construction of a correspondence between the representation of a binary
quadratic form by the adjoint of a ternary quadratic form and the
representation of the determinant of the binary form by the ternary form. This
led him to his famous result expressing the number of representations of an
integer as a sum of 3 squares in term of class numbers. Inverting
$\alpha_{n,n-1}$ is algorithmically the computation of the kernel of a rank
one integral matrix which may be effected by the Hermite normal form algorithm
(see section 2.2).

In general $G_{n,k}({\mathbb{Z}})=Im(\alpha_{n,k})$ is only a proper subset of
$\wedge^{k}{\mathbb{Z}}^{n}$ when $1<k<n-1$, but a simple parameter count (see
comment after corollary 3.5) shows that $G_{n,k}({\mathbb{Z}})$ lies inside a
subvariety of codimension $d_{n,k}:=\begin{pmatrix}n\cr
k\end{pmatrix}-(kn-(k^{2}-1))$ of $\wedge^{k}{\mathbb{R}}^{n}$. This is just
the codimension of the Grassmannian in its Plücker embedding. Clearly,
$d_{n,k}=d_{n,n-k}=0$ for $k=1,n-1$ and $d_{n,k}=1$ occurs only once for
$n=4,k=2$. This is perhaps the most interesting case since the single relation
allows one to impose a group law, and this is exactly Gauss’s composition law
for binary quadratic forms or more generally Bhargava’s [1] composition law
for ${\mathbb{Z}}^{2}\otimes{\mathbb{Z}}^{2}\otimes{\mathbb{Z}}^{2}$ cubes.
Indeed the invariance of the discriminants of the binary forms
$Q_{i}^{A},i=1,2,3$ attached to a Bhargava’s $2\times 2\times 2$ cube $A$ is
equivalent to $Im(\alpha_{4,2})$ lying on the Pfaffian subvariety

(1.4) $X_{12}X_{34}-X_{13}X_{24}+X_{14}X_{23}=0,$

of ${\mathbb{Z}}^{6}$.

So the algorithmic problem of given any two binary quadratic forms
$Q_{i}(x,y),i\in\\{2,3\\}$ of the same discriminant, to find a Bhargava’s cube
$A$ so that $Q_{i}^{A}=Q_{i}$ (see section 3.2) is equivalent to inverting
$\alpha_{4,2}$. Both D. Shanks (see [9] F. Lemmermeyer ) and [11] Alf van der
Poorten have considered this algorithmic problem and [11] also observed that
it is equivalent to inverting $\alpha_{4,2}$. In section 3.1 (Theorem 3.2), we
give an explicit algorithm for inverting $\alpha_{n,2}$ which involves only a
single GCD calculation and for general $n$, and this appears to be new. By
specializing, it gives rise to an explicit composition law for
${\mathbb{Z}}^{2}\otimes{\mathbb{Z}}^{2}\otimes{\mathbb{Z}}^{2}$ cube and in
particular recovers an old explicit rule for composing binary quadratic forms
given by Arndt (see Theorem 3.3).

As Bhargava [1] showed, the above gives an algorithm for composing $2\times
2\times 2$ cubes (Theorem 2 [1]) which additionally gives rise to algorithms
for four new composition laws.

In section 4, we show that Gauss’s argument on inverting $\alpha_{3,2}$ may be
generalized so that any given symmetric positive definite matrix $A$ induces a
natural metric on $G_{n,k}({\mathbb{Z}})$ so that the map $X\rightarrow
X^{T}AX$ becomes norm preserving, in the sense that an $X$ of $\hat{A}$ norm
$m$ is mapped to a $k-ary$ form of determinant $m$.

##  2\. Duality and Inverting $\alpha_{n,n-1}$

###  2.1. Notations and Duality

We will write $\wedge^{k}{\mathbb{Z}}^{n}$ for the products of
$\begin{pmatrix}n\cr k\end{pmatrix}$ copies of ${\mathbb{Z}}$ indexed by
$I=i_{1},\ldots,i_{k}$ with $1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n$ and we let
$|I|=k$, $||I||=i_{1}+\cdots+i_{k}$. We also write
$G_{n,k}({\mathbb{Z}}):=Im(\alpha_{n,k})\subset\wedge^{k}{\mathbb{Z}}^{n}$ for
the subset of vectors expressible as $Y=v_{1}\wedge\cdots\wedge v_{k}$.

There is a map

$\wedge^{k}{\mathbb{Z}}^{n}\rightarrow\wedge^{n-k}{\mathbb{Z}}^{n}:Y\rightarrow\hat{Y},\;\;\;\hat{Y}_{I}=(-1)^{||\hat{I}||}Y_{\hat{I}},$

where for any index $I$, $\hat{I}=J=j_{1}\ldots j_{n-k}$ is the dual index
with $I\cup J=\\{1,\ldots,n\\}$ and $j_{i}$ increasing. Note that
$\hat{\hat{I}}=I$ but $\hat{\hat{Y}}=(-1)^{(n(n+1))/2}Y$. We also write
$gcd(Y)$ for $gcd\\{Y_{I}:|I|=k\\}$. We will always identify an
$X=(v_{1},\ldots,v_{k})\in({\mathbb{Z}}^{n})^{k}$ with the $n\times k$ matrix
$X=[v_{1}\cdots v_{k}]$ and we will write $X_{I}$ for its $I$-th Plücker
coordinate as given in (1.1). We will say that $X$ is primitive system if
$gcd_{I}(X_{I})=1$ and we note that this is equivalent to $\alpha_{n,k}(X)$ is
a primitive vector in $\wedge^{k}{\mathbb{Z}}^{n}$ in the usual sense. Note
that a necessary and sufficient condition for primitivity of $X$ is that it
should be extendable to a $GL_{n}({\mathbb{Z}})$ matrix (see [12]).

By the Cauchy-Binet identity [4], we have a semidefinite pairing on
$({\mathbb{Z}}^{n})^{k}$ given by

(2.1) $<X,Y>:=\det(X^{T}Y)=\sum_{|I|=k}X_{I}Y_{I}.$

The semidefiniteness follows from a Pythagoras theorem :

$||X||^{2}:=\det(X^{T}X)=\sum_{|I|=k}(X_{I})^{2},$

which says that the square of the volume of the parallelepiped spanned by
$v_{1},\ldots,v_{k}$ in ${\mathbb{R}}^{n}$ is the sum of squares of the volume
of its projections on the coordinate $k$\- planes. Moreover we have

$<X,\hat{Y}>=<\hat{X},Y>$

for $X\in({\mathbb{Z}}^{n})^{k},Y\in({\mathbb{Z}}^{n})^{n-k}$, though there is
no linearity so that $<.,.>$ is only a semi-metric.

If in (2.1), $\alpha_{n,k}(X)=v_{1}\wedge\cdots\wedge v_{k}\in
G_{n,k}({\mathbb{Z}})$ and $\alpha_{n,n-k}(Y)=w_{1}\wedge w_{2}\cdots\wedge
w_{n-k}\in G_{n,k}({\mathbb{Z}}),$ are decomposable vectors of complementary
dimensions then

(2.2) $<X,\hat{Y}>=\det([v_{1},\ldots,v_{k},w_{1},\ldots,w_{n-k}])=X\wedge Y,$

by the generalized column expansion of determinant.

###  2.2. Inverting $\alpha_{n,n-1}$

We will consider algorithms for inverting $\alpha_{n,k}$, by which we mean
finding any one of its pre-images. Note that by Lemma 3.4 we can get all the
images via an $SL_{k}$ transformation once we have found one.

We consider first the co-dimension one case. Let
$v_{1},\dots,v_{n-1}\in{\mathbb{Z}}^{n}$, by (2.2),

$<v_{1}\wedge\cdots\wedge v_{n-1},\hat{v_{j}}>=\det([v_{1},\cdots
v_{n-1},v_{j}])=0,$

for $1\leq j\leq n-1$. This says $v_{1}\wedge\cdots\wedge v_{n-1}$ is
orthogonal to all the $v_{j}$ . So given $0\neq
v\in\wedge^{n-1}{\mathbb{Z}}^{n}\cong{\mathbb{Z}}^{n},gcd(v)=1$, the algorithm
for inverting $\alpha_{n,n-1}$ is to first find an integral basis
$\\{v_{1},\ldots,v_{n-1}\\}$ for the rank-one system $v^{T}x=0$ and we will
have $\alpha_{n,n-1}(v_{1}\wedge\cdots\wedge v_{n-1})=v$. If $gcd(v)=\lambda$,
one simply replaces $v_{1}$ by $\lambda v_{1}$. So the problem reduces to
computing the kernel of a single row integral matrix.

Computing a basis of the kernel of an integral matrix $A$ can be effected by
the well known algorithm for computing Hermite normal form. We transform $A$
to HNF : $H=AU$ where $U\in GL_{n}({\mathbb{Z}})$, then the columns of $U$
corresponding to zero columns of $H$ gives a basis (see for example, [3]
Proposition 2.4.9). In PARI, all information can be read off by a single
command $a=mathnf(A,2)$.

The above algorithm for inverting $\alpha_{n,n-1}$ should be viewed as an
algorithm for inverting $\alpha_{n,n-k}$ given that one can invert
$\alpha_{n,k}$ as given by the next lemma .

######  Lemma 2.1.

Let $0<k<n-k$ and given $w_{1},\cdots,w_{k}\in{\mathbb{Z}}^{n}$ such that
$W=w_{1}\wedge\cdots\wedge w_{k}\in\wedge^{k}({\mathbb{Z}}^{n})$ is not the
zero vector. Then one can find vectors
$v_{1},\cdots,v_{n-k}\in{\mathbb{Z}}^{n}$ with $v_{1}\wedge\cdots\wedge
v_{n-k}=\hat{W}$.

###### Proof.

We can clearly assume $gcd(W)=1$ as before. We compute a basis
$v_{1},\ldots,v_{n-k}$ for the kernel of the $k\times n$ matrix with the
$w_{i}$ as rows by a HNF algorithm so that $w_{i}\cdot
v_{j}=0,i=1,\ldots,k,j=1,\ldots,n-k,\;gcd(v_{1}\wedge\cdots\wedge v_{n-k})=1$.
By (2.2), both $\hat{W}$ and $V=v_{1}\wedge...\wedge v_{n-k}$ are orthogonal
to the adjoint of

$v_{i_{1}}\wedge\cdots\wedge v_{i_{k}},\;(v_{i_{1}}\wedge\cdots\wedge
v_{i_{k-1}})\wedge w_{j_{1}},\;(v_{i_{1}}\wedge\cdots\wedge
v_{i_{k-2}})\wedge(w_{j_{1}}\wedge
w_{j_{2}}),\cdots,v_{i_{1}}\wedge(w_{j_{1}}\wedge\cdots\wedge w_{j_{k-1}})$

which spans a space of codimension $\begin{pmatrix}n\cr
n-k\end{pmatrix}-\sum_{j=0}^{k-1}\begin{pmatrix}n-k\cr
k-j\end{pmatrix}\begin{pmatrix}k\cr j\end{pmatrix}=1$ in
$\wedge^{k}{\mathbb{R}}^{n}$ so that $\hat{W}=\pm v_{1}\wedge\cdots\wedge
v_{n-k}$ since both have components with gcd 1. ∎

Since inverting $\alpha_{n,1}$ is trivial, we get the algorithm for inverting
$\alpha_{n,n-1}$ above. For $k>1$ one needs to first determine which points in
$\wedge^{k}{\mathbb{Z}}^{n}$ are decomposable and a separate algorithm is
needed. We will do this first for $k=2$ which tells us how to generalize to
arbitrary $k$.

##  3\. Inverting $\alpha_{n,2}$ and constructing Bhargava’s cube

###  3.1. Inverting $\alpha_{n,2}$

If
$x=(x_{1},x_{2},x_{3},x_{4})^{T},y=(y_{1},y_{2},y_{3},y_{4})^{T}\in{\mathbb{Z}}^{4}$
and $X=x\wedge y\in{\mathbb{Z}}^{6}$, a simple calculation shows that

$X_{12}X_{34}+X_{14}X_{23}=X_{13}X_{24}.$

It follows that $G_{n,2}({\mathbb{Z}})$ must lie in the intersections given by
the Plücker relations

(3.1) $X_{ij}X_{kl}-X_{ik}X_{jl}+X_{il}X_{jk}=0,\;\;\;1\leq i<j<k<l\leq n,$

of $\wedge^{2}{\mathbb{Z}}^{n}$. There are $\begin{pmatrix}n\cr
4\end{pmatrix}$ relations in (3.1), but they are clearly not independent.
Given any non-zero $X$ (say with $X_{12}\neq 0)$, we will show that (theorem
3.2 below) we can find $x,y\in{\mathbb{Z}}^{n}$ with $x\wedge y=X$ using only
those relations where $X_{12}$ occurs :

(3.2) $X_{12}X_{kl}-X_{1k}X_{2l}+X_{1l}X_{2k}=0,\;\;\;3\leq k<l\leq n.$

We note that this gives exactly $\begin{pmatrix}n-2\cr 2\end{pmatrix}$
independent relations, which agree with the codimension $d_{n,2}$.

We will need first a Lemma from D. Cox ([5], Lemma 3.5) who may have
anticipated the geometric view as he stated his result in the exact
determinantal form that we need.

######  Lemma 3.1.

Let $p=(p_{1},\ldots,p_{r})^{T},q=(q_{1},\ldots,q_{r})^{T}\in{\mathbb{Z}}^{r}$
and $y$ be an integer such that $gcd(y,p_{1},\ldots,p_{r})=1$. Then the
congruence $px\equiv q$ mod $y$ has a unique solution mod $y$ if and only if
$p\wedge q=0$ mod $y$.

We note that moreover $x$ in the above is easily computable by a single GCD
computation. Let $L\in{\mathbb{Z}}^{r},\ell\in{\mathbb{Z}}$ be such that
$L\cdot p+\ell y=1$. Then $L\cdot p\equiv 1$ mod $y$, so we have

(3.3) $x\equiv L\cdot q\;\;mod\;\;y.$

######  Theorem 3.2.

The Plücker relations (3.1) defined the integral Grassmannian
$G_{n,2}({\mathbb{Z}})=Im(\alpha_{n,2})$ in the Plücker embedding in
$\wedge^{2}{\mathbb{Z}}^{n}$. There is an explicit algorithm (involving only a
single GCD computation) for finding a pre-image of any $0\neq X\in
G_{n,2}({\mathbb{Z}})$ using only $d_{n,2}$ of the relations containing a
particular $X_{ij}$ where $X_{ij}\neq 0$.

###### Proof.

Let $X\in Im(\alpha_{n,2})$, the relations (3.1) are clearly necessary.
Without loss of generality we may assume that $X_{12}\neq 0$, and we will find
$x,y\in{\mathbb{Z}}^{n}$ with $x\wedge y=X$. By the $SL_{2}({\mathbb{Z}})$
invariance, if we have a solution, we can bring the first two rows of $[x,y]$
into Hermite normal form, so we can expect to find $x,y$ with
$y_{1}=0,x_{1}y_{2}=X_{12}$ and $|x_{2}|<|y_{2}|$ which leads us to the
following algorithm.

We first compute integers $\lambda_{j}$ (and this is the only computation
needed) such that

(3.4) $x_{1}=gcd(X_{12},\ldots,X_{1n})=\sum_{j=2}^{n}\lambda_{j}X_{1j},$

and set

$y_{1}=0,\;\;\;y_{j}=\frac{X_{1j}}{x_{1}},j=2,\ldots,n.$

We then let $p=(y_{3},\ldots
y_{n})^{T},q=(X_{23},\ldots,X_{2n})\in{\mathbb{Z}}^{n-2}$. Since
$gcd(y_{2},y_{3},\ldots,y_{n})=gcd(X_{12}/x_{1},\ldots,X_{1n}/x_{1})=1$, and
by (3.2) for $3\leq k<l\leq n$,

$p_{k}q_{l}-q_{k}p_{l}=\frac{X_{1k}X_{2l}-X_{1l}X_{2k}}{x_{1}}=\frac{-X_{12}X_{kl}}{x_{1}}=-y_{2}X_{kl},$

so we have $p\wedge q\equiv 0$ mod $y_{2}$. By Lemma 3.1 there is a unique
$x_{2}$ mod $y_{2}$ such that

$x_{2}y_{j}\equiv X_{2j}\;\;mod\;\;y_{2},j=3,\ldots n,$

which by (3.3) and the above, is given explicitly by

(3.5) $x_{2}=\sum_{j=3}^{n}\lambda_{j}X_{2j}\;\;mod\;\;y_{2},$

where the $\lambda_{j}$ is computed earlier in (3.4) so that

(3.6)
$x_{j}:=\frac{x_{2}y_{j}-X_{2j}}{y_{2}}\in{\mathbb{Z}},\;\;j=3,\ldots,n.$

We have thus found vectors $x,y\in{\mathbb{Z}}^{n}$ with

(3.7) $X_{ij}=x_{i}y_{j}-x_{j}y_{i},1\leq i<j\leq n,i\leq 2.$

It then follows from (3.2) and (3.7) that for $2<i<j\leq n$, we have

$\displaystyle X_{ij}$ $\displaystyle=$
$\displaystyle\frac{X_{1i}X_{2j}-X_{1j}X_{2i}}{X_{12}}$ $\displaystyle=$
$\displaystyle\frac{(x_{1}y_{i}-y_{1}x_{i})(x_{2}y_{j}-y_{2}x_{j})-(x_{1}y_{j}-y_{1}x_{j})(x_{2}y_{i}-y_{2}x_{i}))}{x_{1}y_{2}-y_{1}x_{2}}=x_{i}y_{j}-y_{i}x_{j}.$

So we have found $x,y$ with $X=x\wedge y$ using only (3.2), but (3.1) must now
holds since each $X_{ij}=x_{i}y_{j}-x_{j}y_{i}$. ∎

###  3.2. Constructing Bhargava’s cube by inverting $\alpha_{4,2}$

We now show via Bhargava’s [1] composition law on $2\times 2\times 2$ cubes
that inverting $\alpha_{4,2}$ is the main algorithmic step in Gauss’s
composition law for binary quadratic forms. This seems to give an interesting
geometric and simple way to view this important algorithm.

Let $A$ be a Bhargava’s [1] $2\times 2\times 2$ cube of integers, which is an
optimally symmetric description. We will fix one of the three (say the
front/back) slicing and let

(3.8) $M_{1}=\begin{pmatrix}x_{1}&x_{2}\cr
x_{3}&x_{4}\end{pmatrix},\;\;N_{1}=\begin{pmatrix}y_{1}&y_{2}\cr
y_{3}&y_{4}\end{pmatrix},$

be the front and back faces. We also have

(3.9) $\displaystyle M_{2}=\begin{pmatrix}x_{1}&x_{3}\cr
y_{1}&y_{3}\end{pmatrix},\;\;$ $\displaystyle
N_{2}=\begin{pmatrix}x_{2}&x_{4}\cr y_{2}&y_{4}\end{pmatrix},$ (3.10)
$\displaystyle M_{3}=\begin{pmatrix}x_{1}&y_{1}\cr
x_{2}&y_{2}\end{pmatrix},\;\;$ $\displaystyle
N_{3}=\begin{pmatrix}x_{3}&y_{3}\cr x_{4}&y_{4}\end{pmatrix},$

corresponding to the left/right, top/bottom faces. The associated binary
quadratic forms are

$Q_{i}^{A}(x,y)=-\det(M_{i}x-N_{i}y),\;\;i=1,2,3,$

and these sum to zero in the class group.

Corresponding to our slicing are two vectors
$x=(x_{1},x_{2},x_{3},x_{4})^{T},y=(y_{1},y_{2},y_{3},y_{4})^{T}\in{\mathbb{Z}}^{4}$
and their wedge product $X=x\wedge y$ with components
$X_{ij}=\det\begin{pmatrix}x_{i}&y_{i}\cr x_{j}&y_{j}\end{pmatrix}$ satisfying
(1.4). A calculation gives

(3.11) $Q_{1}^{A}=(-\det M_{1},\det\begin{pmatrix}x_{1}&x_{2}\cr
y_{3}&y_{4}\end{pmatrix}-\det\begin{pmatrix}x_{3}&x_{4}\cr
y_{1}&y_{2}\end{pmatrix},-\det N_{1}),$ (3.12) $\displaystyle Q_{2}^{A}$
$\displaystyle=$ $\displaystyle(-X_{13},X_{14}+X_{23},-X_{24}),$ (3.13)
$\displaystyle Q_{3}^{A}$ $\displaystyle=$
$\displaystyle(-X_{12},X_{14}-X_{23},-X_{34}),$

and we note the invariance of $Disc(Q_{i}^{A})$ is equivalent to (1.4).

Suppose now we are given two primitive binary quadratic forms
$Q_{2}=(a_{2},b_{2},c_{2}),Q_{3}=(a_{3},b_{3},c_{3})$ of the same
discriminant, then the vector

(3.14) $X=\begin{pmatrix}X_{12}\cr X_{13}\cr X_{14}\cr X_{23}\cr X_{24}\cr
X_{34}\end{pmatrix}:=\begin{pmatrix}-a_{3}\cr-
a_{2}\cr(b_{2}+b_{3})/2\cr(b_{2}-b_{3})/2\cr-c_{2}\cr-c_{3}\end{pmatrix}\in
Im(\alpha_{4,2}),$

since $X$ satisfies (1.4) by the equality of the discriminants. Furthermore by
the primitivity of $Q_{2},Q_{3}$, $X$ is primitive ($gcd(X)=1$). By our
algorithm for inverting $\alpha_{4,2}$ , we can find $x,y\in{\mathbb{Z}}^{4}$
with $x\wedge y=X$. We can now construct a $2\times 2\times 2$ cube $A$ with
$x,y$ as the front/back face and we get $Q_{i}^{A}=Q_{i},i=2,3$. so we have
found $[Q_{1}^{A}]=-([Q_{2}]+[Q_{3}])$ in the class group as Bhargava had
shown. Also by section 3.3 below $\alpha_{4,2}^{*}$ is injective, so $x,y$ and
hence $Q_{1}^{A}$ is determined up to $SL_{2}({\mathbb{Z}})$ transformation,
so that there is a unique composed class. By applying the explicit algorithm
for inverting $\alpha_{4,2}$ in Theorem 3.2 to the vector $X$, we obtain the
following simple algorithm for constructing Bhargava’s cube which leads to an
explicit algorithm for Gauss’s composition agreeing with that given in
([2],Theorem 4.10) attributed to Arndt.

######  Theorem 3.3.

Given two primitive binary quadratic forms
$Q_{2}=(a_{2},b_{2},c_{2}),Q_{3}=(a_{3},b_{3},c_{3})$ of the same discriminant
$D$. Let

(3.15) $\displaystyle x_{1}$ $\displaystyle=$ $\displaystyle
gcd(-a_{3},-a_{2},(b_{2}+b_{3})/2)$ (3.16) $\displaystyle=$
$\displaystyle\lambda_{1}(-a_{3})+\lambda_{2}(-a_{2})+\lambda_{3}(b_{2}+b_{3})/2,\;\;\;\lambda_{i}\in{\mathbb{Z}}$
(3.17) $\displaystyle x_{2}$ $\displaystyle=$
$\displaystyle\lambda_{2}(b_{2}-b_{3})/2-\lambda_{3}c_{2},$

then under Gauss’ composition, a representative of $[Q_{2}+Q_{3}]$ is given by
the primitive form

(3.18)
$Q_{1}^{\prime}=(c_{1},b_{1},a_{1}):=\left(\frac{a_{2}a_{3}}{x_{1}^{2}},b_{2}+2a_{2}\frac{x_{2}}{x_{1}},\frac{(b_{1}^{2}-D)x_{1}^{2}}{4a_{2}a_{3}}\right),$

which specializes to Dirichlet composition ([5] Proposition 3.8) when
$x_{1}=1.$ More over if we set

(3.19) $\displaystyle x_{3}$ $\displaystyle=$
$\displaystyle(2a_{2}x_{2}+(b_{2}-b_{3})x_{1})/(2a_{3})$ (3.20) $\displaystyle
x_{4}$ $\displaystyle=$
$\displaystyle(2c_{2}x_{1}+(b_{2}+b_{3})x_{2})/(-2a_{3})$ (3.21)
$\displaystyle y_{1}$ $\displaystyle=$ $\displaystyle 0$ (3.22) $\displaystyle
y_{2}$ $\displaystyle=$ $\displaystyle-a_{3}/x_{1}$ (3.23) $\displaystyle
y_{3}$ $\displaystyle=$ $\displaystyle-a_{2}/x_{1}$ (3.24) $\displaystyle
y_{4}$ $\displaystyle=$ $\displaystyle(b_{2}+b_{3})/(2x_{1}),$

and let $A$ be the Bhargava’s cube with front and back faces given by (3.8).
Then $Q_{i}^{A}=Q_{i},i=2,3$. The algorithm involves only a single GCD
computation.

###### Proof.

By applying the algorithm in Theorem 3.2 to $X\in Im(\alpha_{4,2})$ given by
(3.12), we find $x,y$ with $x\wedge y=X$ from (3.4),(3.5),(3.6) which gives
(3.13), (3.15) and the cube $A$ with $Q_{1}=(a_{1},b_{1},c_{1})$ from (3.10)
where $a_{1},b_{1},c_{1}$ are as defined in (3.14). Since
$[Q_{1}]=-[Q_{2}+Q_{3}]$, we get $-Q_{1}=(a_{1},-b_{1},c_{1})$ for the
composed class $[Q_{2}+Q_{3}]$, and applying the $SL_{2}({\mathbb{Z}})$
transformation $x^{\prime}=-y,y^{\prime}=x$ to $-Q_{1}$ gives the more
standard equivalent form $Q_{1}^{\prime}=(c_{1},b_{1},a_{1})$ in (3.14). The
only algorithmic step is (3.13). ∎

###  3.3. Further composition laws on cubes

The construction above also solves the algorithmic problem of finding the
composition of two Bhargava’s projective cubes ([1] Theorem 2). Given two
projective cubes $A,A^{\prime}$ we can compute by the above primitive forms
$Q_{2},Q_{3}$ so that

$[Q_{2}]=[Q_{2}^{A}]+[Q_{2}^{A^{\prime}}],[Q_{3}]=[Q_{3}^{A}]+[Q_{3}^{A^{\prime}}],$

we can then find a cube $A^{\prime\prime}$ with
$Q_{i}^{A^{\prime\prime}}=Q_{i},i=2,3$ by the above. We then have

$\displaystyle[Q_{1}^{A^{\prime\prime}}]$ $\displaystyle=$
$\displaystyle-([Q_{2}^{A^{\prime\prime}}]+[Q_{3}^{A^{\prime\prime}}])=-([Q_{2}]+[Q_{3}])$
$\displaystyle=$
$\displaystyle-([Q_{2}^{A}]+[Q_{2}^{A^{\prime}}]+[Q_{3}^{A}]+[Q_{3}^{A^{\prime}}])=([Q_{1}^{A}]+[Q_{1}^{A^{\prime}}]),$

so $[Q_{i}^{A^{\prime\prime}}]=[Q_{i}^{A}]+[Q_{i}^{A^{\prime}}],i=1,2,3$, and
hence $[A^{\prime\prime}]=[A]+[A^{\prime}]$ and $[A^{\prime\prime}]$ is
projective since composition of primitive forms is primitive. As Bhargava
showed in [1], the composition law on cubes gives rise to 4 further
composition laws and hence the algorithm extends to these also.

###  3.4. $\alpha_{n,k}^{*}$ is injective

In ([6] Art 234), Gauss proved that if $a,b,c,d$ are vectors in
${\mathbb{Z}}^{n}$ with $c\wedge d=t(a\wedge b),t\in{\mathbb{Z}}$ and
$gcd(a\wedge b)=1,$ then $[c,d]=[a,b]H,$ where $H\in
GL_{2}({\mathbb{Z}}),\det(H)=t$. In particular this implies that
$\alpha_{n,2}^{*}$ restricted to primitive system of vectors is injective.
Gauss’ proof can be generalized and the matrix $H$ can be given explicitly. We
give here the proof for the case $k=3$ and the explicit form of $H$ which
makes it clear that the result holds for all $k$. We are indebted to the
referees of ANTS VIII for the following which greatly simplifies our earlier
proof.

######  Lemma 3.4.

(case $k=3$) Let $a,b,c,e,f,g$ be vectors in ${\mathbb{Z}}^{n}$ and $0\neq
t\in{\mathbb{Z}}$ with $e\wedge f\wedge g=t(a\wedge b\wedge c),$ and
$gcd(a\wedge b\wedge c)=1$. Let $L\in{\mathbb{Z}}^{n}$ be such that
$L\cdot(a\wedge b\wedge c)=1$, and let $H$ be the following integral matrix
(where $\cdot$ denotes inner product)

$H=L\cdot\begin{pmatrix}e\wedge b\wedge c&f\wedge b\wedge c&g\wedge b\wedge
c\cr a\wedge e\wedge c&a\wedge f\wedge c&a\wedge g\wedge c\cr a\wedge b\wedge
e&a\wedge b\wedge f&a\wedge b\wedge g\end{pmatrix},$

then we have

$[a,b,c]H=[e,f,g],\;\;\;\;\det H=t.$

###### Proof.

We need to show that

(3.25) $L\cdot(e\wedge b\wedge c)a+L\cdot(a\wedge e\wedge c)b+L\cdot(a\wedge
b\wedge e)c=e,$

for the first entry and the rest follows similarly. Clearly, $a\wedge b\wedge
c\wedge e=0$, so we know $a,b,c,e$ are linearly dependent. But since we assume
that $gcd(a\wedge b\wedge c)=1$, $a,b,c$ are linearly independent, so we have
$e=\lambda_{1}a+\lambda_{2}b+\lambda_{3}c$, for $\lambda_{i}\in{\mathbb{R}}$.
It follows that $L\cdot(e\wedge b\wedge c)=L\cdot(\lambda_{1}a\wedge b\wedge
c)=\lambda_{1}\in{\mathbb{Z}},$ and likewise for $\lambda_{2},\lambda_{3}$. ∎

The lemma implies the following :

######  Corollary 3.5.

The wedge map $\alpha_{n,k}$ gives a bijection

(3.26)
$({\mathbb{Z}}^{n}-0)^{k*}/SL_{k}({\mathbb{Z}})\cong(G_{n,k}({\mathbb{Z}}))^{*},$

where $*$ denotes restriction to nonzero primitive system of vectors.

(3.17) implies a simple counting argument which gives our formula for
$d_{n,k}$ in the introduction. We first note that restriction to primitive
system of vectors does not reduce dimension. Thinking of $\alpha_{n,k}$ in
(1.3) as a covering map mapping ${\mathbb{Z}}^{kn}$ onto
$G_{n,k}({\mathbb{Z}})$ with ”fiber” $SL_{k}({\mathbb{Z}})$ depending on
$k^{2}-1$ parameters gives us $kn-(k^{2}-1)$ free parameters for
$G_{n,k}({\mathbb{Z}})$, and hence the formula for $d_{n,k}$.

##  4\. Inverting $\alpha_{n,k}$ and representations of forms

In ([6] Art.279), Gauss gave an algorithm for inverting $\alpha_{3,2}$. It is
interesting to note that he was concerned here with the representation of
integers by ternary form, rather than his composition law. We generalized his
argument here which gives one reason why one may wish to invert
$\alpha_{n,k}$.

Let $A$ be a fixed matrix for an $n$-ary form and
$X\in({\mathbb{Z}}^{n})^{k}$, then the well known map $\phi_{A}:X\rightarrow
X^{T}AX$ transforms $A$ to a $k$-ary form. Two different $X$ define equivalent
$k$-ary form if and only if they differ by an $SL_{k}({\mathbb{Z}})$
transform, so one should think of $X$ as lying in the integral Grassmannian
$G_{n,k}({\mathbb{Z}})$. Furthermore, as we shall show, $A$ induces a natural
metric on $G_{n,k}({\mathbb{Z}})$ so that $\phi_{A}$ becomes volume
preserving.

For $X=[v_{1},...,v_{k}],Y=[w_{1},...,w_{k}]\in({\mathbb{Z}}^{n})^{k}$, (2.1)
gives a ”flat” semi-definite pairing. The matrix $A$ induces correspondingly a
semi-definite pairing on $({\mathbb{Z}}^{n})^{k}$, given by

(4.1) $<X,Y>_{A}:=\det(X^{T}AY)=\sum_{|I|,|J|=k}\hat{A}_{IJ}X_{I}Y_{J},$

where $\hat{A}$ is the $k$-skew symmetric adjoint of $A$ with $IJ$ component
given by

(4.2) $\hat{A}_{IJ}=\det\begin{pmatrix}A_{i_{1}j_{1}}&\cdots
A_{i_{1},j_{k}}\cr...&...\cr A_{i_{k},j_{1}}&\cdots
A_{i_{k}j_{k}}\end{pmatrix}.$

(4.1) follows from applying the Cauchy-Binet identity (2.1) twice, and hence
it is an algebraic identity which holds even when $A$ is not necessarily
symmetric or positive definite.

Putting $Y=X$, (4.1) says that the representation of a $k$-ary form

(4.3) $\phi_{k}=X^{T}AX$

by an $n$-ary form $A$ induces a representation of the integer $\det\phi_{k}$
by the $n_{k}$-ary adjoint form $\hat{A}$ of $A$,

(4.4) $\det\phi_{k}=x^{T}\hat{A}x,$

by setting for $X=[v_{1},...,v_{k}]$,

(4.5) $x=v_{1}\wedge...\wedge v_{k}.$

However passing conversely from (4.4) to (4.3) is only possible when one can
invert $\alpha_{n,k}$ at $x$ in (4.5), which also requires $x\in
Im(\alpha_{n,k})=G_{n,k}({\mathbb{Z}})$. Theorem 3.2 gives an algorithm to do
this when $k=2$ and Lemma 2.1 gives the algorithm for the codimension one case
where $\alpha_{n,n-1}$ is always surjective.

It seems interesting to note that the above can be viewed geometrically as
giving a natural $\hat{A}$-norm on the integral Grassmannian
$G_{n,k}({\mathbb{Z}})$ so that $\phi_{A}$ maps vectors of norm $m$ into
volume $m$ $k$-dimensional sublattices of the integral lattice $\Gamma_{A}$
with Gram matrix $A$.

######  Lemma 4.1.

Let $A$ be an integral symmetric positive definite $n\times n$ matrix and
$\Gamma_{A}$ the (classical) integral lattice with Gram matrix $A$. Let
$\hat{A}$ be the $k$-adjoint of $A$ as given in (4.2) which induces a natural
metric on the integral Grassmannian $G_{n,k}({\mathbb{Z}})$. Let
$G_{n,k}({\mathbb{Z}})^{*}_{\hat{A},m}=\\{x:x^{T}\hat{A}x=m,gcd(x)=1\\}$ be
the primitive vectors in $G_{n,k}({\mathbb{Z}})$ with $\hat{A}$-norm
$||x||^{2}_{\hat{A}}:=x^{T}\hat{A}x=m$. Let
$\Gamma^{*}_{A,k,m}=\\{[X^{T}AX]_{SL_{k}({\mathbb{Z}})}:\det(X^{T}AX)=m,gcd(X)=1\\}$,be
the isomorphism classes of primitive $k$ dimensional sublattices of
$\Gamma_{A}$ of determinant m. Then there is a natural map

$\phi_{A,k,m}:G_{n,k}({\mathbb{Z}})^{*}_{\hat{A},m}\rightarrow\Gamma^{*}_{A,k,m}$

mapping

$x=v_{1}\wedge\cdots\wedge v_{k}\rightarrow[X^{T}AX].$

We also note that there is an obvious injection

$\Gamma^{*}_{A,k,m}\subset H_{k}(m)=\\{[\phi_{k}]:\det(\phi_{k})=m\\},$

and it is onto if and only if every classical integral $k$-dimensional
primitive lattice of determinant $m$ can be embedded as a sublattice of
$\Gamma_{A}$. This is known to be true for example for $A=I,k\leq 5$ and
$n\geq k+3$ [8, 10], by Chao Ko and Mordell, the case $k=1,n=4$ being Lagrange
theorem that every integer is a sum of 4 four squares.

##  5\. Acknowledgements

This paper was first reviewed for ANTS VIII. We are very much indebted to the
referees of (ANTS VIII and IX) for pointing out an error in our initial
version of Theorem 3.2, and for simplifying the proof of Lemma 3.4, as well as
giving many useful comments and suggestions, which greatly improve our
presentations. Some of our results were found while preparing a talk at the
Max-Planck-Institut für Mathematik in July 2008. We thank the institute for
hospitality.

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