# Noncommutative Poisson structure and invariants of matrices

F. Eshmatov  ,  X. García-Martínez  and  R. Turdibaev<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>New Uzbekistan
University, 100007 Tashkent, Uzbekistan — Central Asian University, National
Park Street 264, 111221, Tashkent, Uzbekistan  CITMAga & Universidade de Vigo,
Departamento de Matemáticas, Esc. Sup. de Enx. Informática, Campus de Ourense,
E–32004 Ourense, Spain  CITMAga & Universidade de Santiago de Compostela,
Departamento de Matemáticas, Rúa Lope Gómez de Marzoa, s/n, 15782 Santiago de
Compostela, Spain

###### Abstract.

We introduce a novel approach that employs techniques from noncommutative
Poisson geometry to comprehend the algebra of invariants of two $n\times n$
matrices. We entirely solve the open problem of computing the algebra of
invariants of two $4\times 4$ matrices. As an application, we derive the
complete description of the invariant commuting variety of $4\times 4$
matrices and the fourth Calogero-Moser space.

###### Key words and phrases:

Noncommutative Poisson structure, invariant theory, commuting variety,
Calogero-Moser space

###### 2020 Mathematics Subject Classification:

16R30, 16S38, 14A22, 13A50

This work was supported by Agencia Estatal de Investigación de España (Spain,
European FEDER support included), grants PID2020-115155GB-I00 and
PID2021-127075NA-I00, and by Xunta de Galicia through the Competitive
Reference Groups (GRC), ED431C 2023/31.

##  1\. Introduction

In the realm of algebraic geometry, there exists a categorical equivalence
between affine schemes and commutative algebras. Many geometric structures on
affine schemes can be algebraically described, and conversely. However, this
correspondence is not applicable to associative algebras.

Kontsevich and Rosenberg proposed a heuristic principle to explore
noncommutative geometry on associative algebras, which may not necessarily be
commutative [23]. This principle can be broadly summarized: for an associative
algebra $A$ over $\mathbb{C}$, any noncommutative geometric structure—such as
noncommutative Poisson, noncommutative symplectic, etc.—should naturally
induce its classical counterpart on the affine variety $\mathrm{Rep}_{n}(A)$
classifying all representations of $A$ on $\mathbb{C}^{n}$, for every
$n\in\mathbb{N}$.

Crawley-Boevey introduced an $H_{0}$-Poisson structure on any associative
algebra [6]. Essentially, it is a bilinear bracket on $A$ which, in turn,
induces a Lie bracket on the commutator quotient space $A_{\natural}\coloneq
A/[A,A]$. This structure aligns with the KR principle. More precisely, let
$\mathbb{C}[\mathrm{Rep}_{n}(A)]$ be the coordinate ring of
$\mathrm{Rep}_{n}(A)$. The natural trace map

$\mathrm{Tr}_{n}(A)\colon
A_{\natural}\to\mathbb{C}[\mathrm{Rep}_{n}(A)]\,,\quad\bar{a}\to[\rho\mapsto\mathrm{Tr}\,\rho(a)]$
(1.1)

sends the elements of $A_{\natural}$ to functions on $\mathrm{Rep}_{n}(A)$.
The image of $\mathrm{Tr}_{n}(A)$ lies in the subalgebra
$\mathbb{C}[\mathrm{Rep}_{n}(A)]^{\mathrm{GL_{n}}}$ which represents the
coordinate ring of the character variety
$\mathrm{Rep}_{n}(A)\mathord{/\mkern-6.0mu/}\mathrm{GL_{n}}$. Then the main
result of [6] states that if $A$ possesses a $H_{0}$-Poisson structure then
there is a unique Poisson bracket on
$\mathbb{C}[\mathrm{Rep}_{n}(A)]^{\mathrm{GL_{n}}}$ so that the trace map
(1.1) is a Lie algebra homomorphism. Equivalently, the algebra homomorphism
induced from (1.1)

$\mathrm{Sym\,{Tr}}_{n}(A)\colon\mathrm{Sym}(A_{\natural})\to\mathbb{C}[\mathrm{Rep}_{n}(A)]^{\mathrm{GL_{n}}}$
(1.2)

is a Poisson homomorphism.

Let $\mathcal{M}_{n}$ be the vector space of $n\times n$ matrices over the
field $\mathbb{C}$. Consider a positive integer $d$ and the action of the
general linear group $\mathrm{GL}_{n}(\mathbb{C})$ on the direct product
$\mathcal{M}_{n}^{d}$ of $d$ copies of $\mathcal{M}_{n}$ by simultaneous
conjugation. This action sends a $d$-tuple $(X_{1},\dots,X_{d})$ to another
$d$-tuple $(gX_{1}g^{-1},\dots,gX_{d}g^{-1})$ for any
$g\in\mathrm{GL}_{n}(\mathbb{C})$. Consequently, this action induces an action
of $\mathrm{GL}_{n}$ on the algebra $\mathbb{C}[\mathcal{M}_{n}^{d}]$,
representing polynomial functions on $\mathcal{M}_{n}^{d}$. One of the
fundamental questions in invariant theory revolves around describing the
generators and relations of the algebra
$C_{nd}\coloneq\mathbb{C}[\mathcal{M}_{n}^{d}]^{\mathrm{GL_{n}}}$. A well-
known result by Procesi [31] and Razmyslov [32] states that $C_{nd}$ is
generated by trace functions and the algebraic relations between them are
derived from the Cayley-Hamilton theorem. Let us provide the following, more
algebraic, interpretation of this result. Consider the free algebra
$R\coloneq\mathbb{C}\langle x_{1},...,x_{d}\rangle$ on $d$ generators. Then
each point $X\coloneq(X_{1},...,X_{d})\in\mathcal{M}_{n}^{d}$ can be uniquely
determined by the representation

$\rho_{X}\colon R\to\mathcal{M}_{n}\,,\quad x_{i}\mapsto X_{i}\,$

and two representations $\rho_{X}$ and $\rho_{Y}$ are isomorphic if and only
if $X$ and $Y$ lie in the same $\mathrm{GL}_{n}$-orbit. Consequently, we have
the equivalence

$C_{nd}\cong\mathbb{C}[\mathrm{Rep}_{n}(R)]^{\mathrm{GL_{n}}}\,,$

and the Procesi-Razmyslov result can be interpreted as the map in (1.2) being
a surjective homomorphism. Thus, if $R$ has an $H_{0}$-Poisson structure, the
map in (1.2) becomes a surjective Poisson homomorphism. Consequently,
describing the algebra $C_{nd}$ is equivalent to providing a description of
the kernel of $\mathrm{Sym\,{Tr}}_{n}(R)$, which forms a Poisson ideal of
$\mathrm{Sym}(R_{\natural})$.

Let us discuss the $H_{0}$-Poisson structure proposed by Kontsevich on the
free algebra with an even number of generators and the corresponding Lie
algebra homomorphism. To this end, we utilize the quiver realization of a free
algebra. Let $Q=(I,H)$ be a finite quiver, where $I$ is the set of vertices
and $H$ is the set of edges. For $\alpha\in\mathbb{N}^{I}$, we define the
space of representations of $Q$ of dimension vector $\alpha$ to be the vector
space of matrices

$\mathrm{Rep}(Q,\alpha)=\bigoplus_{h\in
H}\mathrm{Mat}(\alpha_{h_{0}}\times\alpha_{h_{1}},\mathbb{C})\,,$

where $h_{0}$ and $h_{1}$ are the head and tail of an edge $h$. The group
$G(\alpha)=\prod\operatorname{GL}_{\alpha_{i}}(\mathbb{C})$ acts on
$\mathrm{Rep}(Q,\alpha)$ by conjugation, and the quotient space
$\mathrm{Rep}(Q,\alpha)/G(\alpha)$ represents isomorphism classes of such
representations. Recall that the double quiver $\overline{Q}$ is defined by
adding a reverse edge $a^{\ast}$ for each $a\in H$. Then there is a natural
identification of $\mathrm{Rep}(\overline{Q},\alpha)$ with the cotangent
bundle $T^{*}\mathrm{Rep}(Q,\alpha)$. The latter has a canonical symplectic
structure, which via the symplectic reduction construction [25], induces a
symplectic structure on
$T^{*}\mathrm{Rep}(Q,\alpha)\mathord{/\mkern-6.0mu/}G(\alpha)$. This, in turn,
induces a Poisson bracket on the algebra
$\mathbb{C}[\mathrm{Rep}(\overline{Q},\alpha)]^{G(\alpha)}$.

In [22], Kontsevich defined an infinite-dimensional Lie algebra structure on
the vector space $\mathbb{C}\overline{Q}_{\natural}$, where
$\mathbb{C}\overline{Q}$ is the path algebra of $\overline{Q}$. He showed that
the map

$\mathrm{Tr}_{\alpha}(Q)\colon\mathbb{C}\overline{Q}_{\natural}\to\mathbb{C}[\mathrm{Rep}(\overline{Q},\alpha)]^{G(\alpha)}$
(1.3)

is a Lie algebra homomorphism for all $\alpha\in\mathbb{N}^{I}$. Let $Q$ be
the quiver with one vertex and $k$ edges-loops. Then, the path algebra
$\mathbb{C}\overline{Q}$ is isomorphic to the free algebra $\mathbb{C}\langle
x_{1},\dots,x_{k},y_{1},\dots,y_{k}\rangle$. The vector space
$V=\mathrm{span}(x_{1},\dots,x_{k},y_{1},\dots,y_{k})$ has a canonical
symplectic form $\omega$:

$\omega(x_{i},x_{j})=\omega(y_{i},y_{j})=0\quad\textrm{and}\quad\omega(x_{i},y_{j})=\delta_{ij}.$

Then, the Kontsevich Lie bracket on $\mathbb{C}\overline{Q}_{\natural}$ is
defined as follows:

$\\{u_{1}\cdots u_{p},v_{1}\cdots
v_{q}\\}=\sum\limits_{\begin{subarray}{c}1\leq i\leq p\\\ 1\leq j\leq
q\end{subarray}}\omega(u_{i},v_{j})u_{i+1}\cdots u_{p}u_{1}\cdots
u_{i-1}v_{j+1}\cdots v_{q}v_{1}\cdots v_{j-1},$ (1.4)

where the elements
$u_{1},\dots,u_{p},v_{1},\dots,v_{q}\in\\{x_{1},...,x_{k},y_{1},...,y_{k}\\}$.
As mentioned above, the map (1.3) allows to compute the Poisson bracket on
$C_{nd}$ for $d=2k$. For instance, the explicit Poisson bracket on $C_{32}$
was described this way by Normatov and Turdibaev [29].

The problem of identifying a minimal generating set of $C_{nd}$ and finding
the exact relations among these generators still remains unsolved. Several
noteworthy results in this area have been attained thus far and explicit
descriptions were found for only $C_{2m}$ for all positive integers $m$,
$C_{32}$ and $C_{33}$. The main method utilized is based on the work of
Abeasis and Pittaluga [1], using the representation theory of symmetric and
general linear groups. They employed this approach in a manner reminiscent of
its application in the theory of PI-algebras. Their method finds the minimal
set of generators as a result of a decomposition of a certain module into
irreducible components.

The next goal was to give the description of $C_{42}$. Drensky and Sadikova
[11] discovered the existence of a minimal generating set whose span forms a
semisimple graded $\operatorname{GL}_{2}$-module, and they studied its
structure. As for the defining relations, it seems that the question remained
largely open. To address this, Drensky and La Scala [10] have devised
algorithms based on the representation theory of $\operatorname{GL}_{2}$ and
conducted calculations using standard Maple functions to determine all
defining relations occurring in degrees 12, 13, and 14.

The present paper provides a comprehensive solution to the problem of
identifying every defining relation with respect to the minimal system of
generators of $C_{42}$ in Drensky and Sadikova [11]. Unlike previous
approaches which were confined to generating relations within the same degree,
our method coming from the noncommutative geometry offers the advantage of a
progression from lower to higher degree relations through the Poisson algebra
structure. Remarkably, the highest degree at which these defining relations
appear is $20$. Nevertheless, our approach expedites the procedure, enabling
the combinatorial computation to finalize at degree 16 with a partial analysis
of bidegree (8, 8) being sufficient. We demonstrate that by utilizing the
Poisson algebra structure, a mere 8 relations suffice to build the associative
ideal of defining relations, which is generated by 105 polynomials. The
insights into how the Poisson structure is effectively harnessed to enhance
our computation are given in Subsection 3.6.

With all the defining relations of $C_{42}$ at our disposal, we are confronted
with the task of understanding them. One perspective in invariant theory
suggests the following useful way of representing an invariant by its Hironaka
decomposition. It is well-known, that $C_{nd}$ is Cohen-Macaulay(see [20] and
[36]). In other words, there exists a set of polynomials, called a homogeneous
system of parameters or primary invariants $p_{1},\dots,p_{k}$, such that

$C_{nd}=\bigoplus_{i=1}^{s}q_{i}\mathbb{C}[p_{1},\dots,p_{k}]$ (1.5)

as a $\mathbb{C}[p_{1},\dots,p_{k}]$-module. The set of polynomials
$\\{q_{1},\dots,q_{s}\\}$ is called the secondary invariants and when combined
with the homogeneous system of parameters, they form a set of fundamental
invariants. The number of primary invariants is known to be $k=(d-1)n^{2}+1$
(see for instance an argument by Teranishi [34]). The decomposition itself has
been largely unknown, even for $C_{42}$, where only the primary invariants
were presented by Teranishi [34]. Drensky and La Scala [10] provided some
degree bounds for the secondary invariants of $C_{42}$, but due to an
incomplete set of defining relations, the decomposition remained elusive. Now,
with the acquisition of all defining relations of $C_{42}$, we are able to
find all the secondary invariants presented in Appendix B and thus, the quest
of searching an answer of what is an invariant of pairs of $4\times 4$
matrices is completed in Subsection 3.10.

As an application of our description, we reproved the results of [13] on the
defining relations of the invariant commuting variety of two $4\times 4$
matrices and the fourth Calogero-Moser space. In fact, this can possibly be
further extended to obtain more descriptions of other subvarieties embedded
into the GIT quotient of pairs of matrices.

Concerning larger matrix sizes, Đoković [30] presented a minimal generating
set consisting of 173 polynomials for $C_{52}$ and computed its Hilbert
series. However, it is important to note that the form of these polynomials
differs from those in [3] and [11], indicating that they might not be well-
suited for expressing simple defining relations. Regarding $C_{62}$, the
current knowledge is limited to the Hilbert series only (see [30]). The
methods developed in our work could potentially find application in resolving
these open problems.

### Acknowledgments

We would like to thank Ivan Shestakov, Efim Zelmanov, Vyacheslav Futorny,
Vladimir Dotsenko, Yura Berest and Xiaojun Chen for their valuable
discussions.

##  2\. Preliminaries

We denote by $\mathcal{M}_{n}$ the $\mathbb{C}$-algebra of $n\times n$
matrices over the field of complex numbers $\mathbb{C}$. Let $d$ be a positive
integer and consider the action of the general linear group
$\mathrm{GL}_{n}(\mathbb{C})$ on the direct product $\mathcal{M}_{n}^{d}$ of
$d$ copies of $\mathcal{M}_{n}$ by simultaneous conjugation sending a
$d$-tuple $(X_{1},\dots,X_{d})$ to a $d$-tuple
$(gX_{1}g^{-1},\dots,gX_{d}g^{-1})$ for any $g\in\mathrm{GL}_{n}(\mathbb{C})$.
This action induces an action of $\mathrm{GL}_{n}$ on the algebra
$\mathbb{C}[\mathcal{M}_{n}^{d}]$ of polynomial functions on
$\mathcal{M}_{n}^{d}$, i.e. for $\varphi\in\mathbb{C}[\mathcal{M}_{n}^{d}]$
and $g\in\mathrm{GL}_{n}$:

$(g\ast\varphi)(X_{1},\dots,X_{d})=\varphi(g^{-1}X_{1}g,\dots,g^{-1}X_{d}g).$

Note that one can identify $\mathcal{M}_{n}^{d}$ with the affine space
$\mathbb{C}^{dn^{2}}$. In general, the description of the GIT-quotient
$\mathcal{M}_{n}^{d}/\\!\\!/\mathrm{GL}_{n}$ is essentially an open problem of
invariant theory.

Given that an algebraic variety $V$ is entirely characterized by its algebra
of polynomial functions $\mathbb{C}[V]$, our objective is to describe the
algebra $\mathbb{C}[\mathcal{M}_{n}^{d}/\\!\\!/\mathrm{GL}_{n}]$. However, by
Mumford [26], we can instead study the isomorphic algebra of
$\mathrm{GL}_{n}$-invariant polynomials

$\mathbb{C}[\mathcal{M}_{n}^{d}]^{\mathrm{GL}_{n}}=\\{\varphi\in\mathbb{C}[\mathcal{M}_{n}^{d}]\mid
g\ast\varphi=\varphi,\,\forall g\in\mathrm{GL}_{n}\\}.$

Traditionally, the algebra of invariants
$\mathbb{C}[\mathcal{M}_{n}^{d}]^{\mathrm{GL}_{n}}$ is denoted by $C_{nd}$.
Conjecturing that invariants of matrices of arbitrary sizes are generated by
finitely many trace functions, Artin [2] showed that this is indeed the case
for invariants of small-sized matrices. Later, the conjecture was
independently proven by Razmyslov [32], Procesi [31], Sibirskiĭ [33] and
Helling [19].

######  Theorem 2.1 ([31]).

The algebra $C_{nd}$ is generated over $\mathbb{C}$ by all traces of products
of generic matrices $\operatorname{Tr}(A_{i_{1}}A_{i_{2}}\cdots A_{i_{j}})$,
with $j\leq 2^{n}-1$.

Another appropriate upper bound on the length of the words inside the trace
function was given by Razmyslov [32] to be $n^{2}$. In fact, it is known that
the exact upper bound is the same number determined by Dubnov-Ivanov-Nagata-
Higman theorem in the context of PI-algebras. The algebra $C_{nd}$ is also
referred to as pure or commutative trace algebra. For further background on
these algebras we refer to Drensky and Formanek [9]. The following result,
often called the second fundamental theorem, affirms that the defining
relations are derived from the Cayley-Hamilton theorem.

######  Theorem 2.2 ([32]).

All defining relations of the pure trace algebra $C_{nd}$ over a field of
characteristic zero are consequences of the Cayley-Hamilton theorem.

###  2.3. Generators and defining relations

Dubnov showed [12] that the only free algebra among pure trace algebras
happens to be the following:

$C_{22}=\mathbb{C}[\operatorname{Tr}(X),\operatorname{Tr}(Y),\operatorname{Tr}(X^{2}),\operatorname{Tr}(XY),\operatorname{Tr}(Y^{2})].$

Moreover, it is known that $C_{23}$ is a polynomial algebra with a single
relation from the work of Sibirskiĭ [33] and Formanek [15]. For $C_{2d}$ with
$d>2$, some polynomial relations appear among the generators. With a choice of
specific traceless generic matrices, Drensky [8] is able to give a better
presentation of the defining relations. Another reason to change to traceless
matrices it due to Procesi [31, Section 5], since there is an isomorphism of
$\mathbb{Z}^{d}$-graded algebras:

$C_{nd}\cong\mathbb{C}[u_{1},u_{2},\dots,u_{d}]\otimes C_{n,d}(0),$

where $C_{n,d}(0)$ is the algebra of $\operatorname{GL}_{n}$-invariant
polynomial functions on the direct product of $d$-tuples of the subspace of
traceless $n\times n$ matrices. In fact, those $u_{i}$’s are the traces of
generic matrices.

For $C_{32}$, Teranishi [34] proves that if the traces of the following ten
words

$X,\,Y,\,X^{2},\,XY,\,Y^{2},\,X^{3},\,X^{2}Y,\,XY^{2},\,Y^{3},\,X^{2}Y^{2}$
(2.1)

are zero, then the value of the trace of any word in generic matrices is also
zero. Applying Hilbert theorem, he argues that $C_{32}$ is integral over the
polynomial subalgebra $A$ in those ten variables and he establishes its
Hironaka decomposition $C_{32}=A\oplus\operatorname{Tr}(XYX^{2}Y^{2})A$.
Clearly, a defining relation containing the square of
$\operatorname{Tr}(XYX^{2}Y^{2})$ is eminent and it appears to be quite
complicated in the work of Nakamoto [27]. Once again, shifting to the
traceless versions

$A\coloneq X-\frac{1}{3}\operatorname{Tr}(X)I_{3},\qquad B\coloneq
Y-\frac{1}{3}\operatorname{Tr}(Y)I_{3}$

of the generic matrices $X$ and $Y$, along with replacing the last two
generators with the traces of $[A,B]^{2}$ and $[A,B]^{3}$, where throughout
the article $[A,B]=AB-BA$, appears to significantly simplify [3] the defining
relation.

In [34, Proposition 5.2] a set of 17 primary invariants were found by
Teranishi for $C_{42}$. Using the fact that $\operatorname{GL}_{2}$ acts via
the standard representation on the two-dimensional space spanned by two
generic matrices, Drensky and Sadikova [11] studied the induced
$\operatorname{GL}_{2}$-action on $C_{42}$. They demonstrated the existence of
a minimal set of generators that spans a semisimple graded
$\operatorname{GL}_{2}$-submodule of $C_{42}$ and list the generators of each
simple component. We provide the set of generators arranged by degrees in
Appendix A. The developed techniques to find the defining relations appear to
be quite computationally complicated yielding only in the discovery of the low
degree relations [10]. Hoge [21] adopted these methods to give a presentation
of $C_{33}$.

###  2.4. Hilbert Series and Hironaka Decomposition

One of the useful tools in this study appears to be the usage of the Hilbert
series—the formal power series $\sum_{i,j\geq 0}\dim V_{i,j}\,t^{i}s^{j}$,
where $C_{n,d}=\oplus_{i,j\geq 0}V_{i,j}$ is considered as a bigraded vector
space. The bigraded Hilbert series of $C_{n2}$ are known to be symmetric
rational functions in two variables and for $n=4$ they were computed by
Teranishi [34] and independently by Berele and Stembridge [4]. For our
purposes, it suffices to consider the simply graded Hilbert series of
$C_{42}$, that is obtained by equalizing the variables, which in its
irreducible form is mentioned in [30, Table 3]:

$\dfrac{\left(1-t^{2}+t^{4}\right)\left(1-t-t^{3}+t^{4}+2t^{5}+t^{6}-t^{7}-t^{9}+t^{10}\right)}{(1-t)^{3}\left(1-t^{2}\right)^{4}\left(1-t^{3}\right)^{5}\left(1-t^{4}\right)^{5}}.$
(2.2)

Following [30], let us rescale (2.2) by
$\left(1+{t}^{2}\right)\left(1+t+{t}^{2}\right)\left(1+{t}^{3}\right)^{2}$ to
obtain a fraction with the following new numerator

$\displaystyle
1+2\,{t}^{5}+2\,{t}^{6}+2\,{t}^{7}+4\,{t}^{8}+4\,{t}^{9}+4\,{t}^{10}+4\,{t}^{11}+2\,{t}^{12}+4\,{t}^{13}$
(2.3)
$\displaystyle+4\,{t}^{14}+4\,{t}^{15}+4\,{t}^{16}+2\,{t}^{17}+2\,{t}^{18}+2\,{t}^{19}+{t}^{24},$

and denominator
$\left(1-t\right)^{2}\left(1-{t}^{2}\right)^{3}\left(1-{t}^{3}\right)^{4}\left(1-{t}^{4}\right)^{6}\left(1-{t}^{6}\right)^{2}$.
This hints at a Hironaka decomposition for $C_{42}$ with 17 primary and 48
secondary invariants of their corresponding degrees. A set of 17 primary
invariants was presented by Teranishi [34] using similar methods he employed
for $C_{32}$. Following the beneficial practice of the usage of traceless
versions, we choose the primary invariants to be the traces of the following:

$\displaystyle X,Y$ $\displaystyle A^{2},AB,B^{2}$ $\displaystyle
A^{3},A^{2}B,AB^{2},B^{3}$ (2.4) $\displaystyle
A^{4},A^{3}B,A^{2}B^{2},AB^{3},B^{4}$
$\displaystyle\frac{1}{2}[A,B]^{2},[A,B]^{2}A^{2},[A,B]^{2}B^{2}.$

Only some bounds on degrees of the secondary invariants were suggested in [10]
as a consequence of obtaining low degree defining relations. In Subsection
3.10 we explain how we obtain the set of secondary invariants arranged by
their degrees in Appendix B, matching the numerator (2.3).

###  2.5. Invariant Commuting Variety and Calogero-Moser spaces

Let us consider the following subspace of
$\mathcal{M}_{n}\times\mathcal{M}_{n}$:

$\\{(X,Y)\in\mathcal{M}_{n}\times\mathcal{M}_{n}\mid\mathrm{rank}([X,Y]+I_{n})=1\\}\,.$

The GIT quotient of this space with respect to the
$\operatorname{GL}_{n}$-action is called the $n$-th Calogero-Moser space, and
it is denoted by $\mathcal{C}_{n}$. The Calogero-Moser spaces were studied in
detail by Wilson [37] and he proved that $\mathcal{C}_{n}$ is a smooth affine
irreducible complex symplectic variety of dimension $2n$. These spaces appear
to be of substantial importance in many areas, such as algebraic geometry
(Hilbert schemes), deformation theory (symplectic reflection algebras),
representation theory (double affine Hecke algebras) and Poisson geometry.
Consequently, there is a strong motivation to obtain a clear and explicit
description of their coordinate rings. Since the coordinate ring of
$\mathcal{C}_{n}$ is a quotient algebra of $C_{n2}$, it is of great interest
to see the refinement of the fundamental theorems on these spaces.

The commuting variety
$\\{(X,Y)\in\mathcal{M}_{n}\times\mathcal{M}_{n}\mid[X,Y]=0\\}$ while being
interesting on its own, its GIT quotient called the $n$-th invariant commuting
variety, which we denote by ${\mathcal{C}om}_{n}$. It is known that
$\mathbb{C}[{\mathcal{C}om}_{n}]$ is isomorphic to $(\mathbb{C}[x,y]^{\otimes
n})^{S_{n}}$ (see [7] and [35]). Calogero-Moser spaces and invariant commuting
varieties appear to share many similarities. It is known that

$\displaystyle\mathbb{C}[\mathcal{C}_{2}]$ $\displaystyle\cong$
$\displaystyle\mathbb{C}[a_{1},a_{2},a_{3},a_{4},a_{5}]/(a_{4}^{2}-a_{3}a_{5}-1),$
$\displaystyle\mathbb{C}[{\mathcal{C}om}_{2}]$ $\displaystyle\cong$
$\displaystyle\mathbb{C}[a_{1},a_{2},a_{3},a_{4},a_{5}]/(a_{4}^{2}-a_{3}a_{5})$

and for $n=3$ the full description is obtained in [28]:

$\mathbb{C}[\mathcal{C}_{3}]\cong\mathbb{C}[a_{1},\dots,a_{9}]/I_{1}\
,\qquad\mathbb{C}[{\mathcal{C}om}_{3}]\cong\mathbb{C}[a_{1},\dots,a_{9}]/I_{0},$

where $I_{\delta}$ is generated by the following five relations:

$\displaystyle r_{1}$ $\displaystyle=a_{3}a_{9}-2a_{4}a_{8}+a_{5}a_{7},$
$\displaystyle r_{2}$ $\displaystyle=a_{5}a_{6}-2a_{4}a_{7}+a_{3}a_{8},$
$\displaystyle r_{3}$ $\displaystyle=9\delta
a_{3}-a_{3}a_{4}^{2}+a_{3}^{2}a_{5}+6a_{6}a_{8}-6a_{7}^{2},$ $\displaystyle
r_{4}$ $\displaystyle=9\delta
a_{4}-a_{4}^{3}+a_{3}a_{4}a_{5}+3a_{6}a_{9}-3a_{7}a_{8},$ $\displaystyle
r_{5}$ $\displaystyle=9\delta
a_{5}-a_{4}^{2}a_{5}+a_{3}a_{5}^{2}+6a_{7}a_{9}-6a_{8}^{2}.$

We refer to Appendix A for the meaning of each of these generators.

Strikingly, the complicated defining relation of $C_{32}$ with the help of the
defining relations of $\mathbb{C}[\mathcal{C}_{3}]$ can be rewritten easily
(see [28]):

$a_{21}^{2}+\frac{4a_{15}^{3}}{27}-\frac{1}{27}(r_{3}r_{5}-r_{4}^{2})-\frac{1}{18}(a_{3}{r_{1}^{2}}-2a_{4}r_{1}r_{2}+a_{5}{r_{2}^{2}})=0.$

For $n=4$ the generators of $\mathbb{C}[\mathcal{C}_{4}]$ and
$\mathbb{C}[{\mathcal{C}om}_{4}]$ are exactly $a_{1},\dots,a_{14}$, since the
other generators of $C_{42}$ contain commutators inside their trace function.
The ideal of the defining relations for the coordinate ring of the fourth
Calogero-Moser space appears to be generated by 12 relations, while for the
invariant commuting variety there are 15 relations [13]. We will see as an
application of knowing the full description of $C_{42}$, how one can deduce
the presentation of $\mathbb{C}[\mathcal{C}_{4}]$ and
$\mathbb{C}[{\mathcal{C}om}_{4}]$ in Section 4.

###  2.6. Poisson algebra structure on $C_{n2}$

Let us consider the symplectic form

$\omega((X_{1},X_{2})\,,\,(Y_{1},Y_{2}))=\textrm{Tr}(X_{1}Y_{2}-X_{2}Y_{1}).$

on the pairs of matrices $\mathcal{M}_{n}\times\mathcal{M}_{n}\cong
T^{*}(\mathcal{M}_{n})$. This form endows
$\mathbb{C}[\mathcal{M}_{n}\times\mathcal{M}_{n}]$ with a Poisson algebra
structure by

$\\{f,g\\}=\omega(X_{f},X_{g}),\quad
f,g\in\mathbb{C}[\mathcal{M}_{n}\times\mathcal{M}_{n}].$

Since the symplectic form is $\operatorname{GL}_{n}$-invariant, this induces
the algebra $C_{n2}$ with a Poisson algebra structure. Nevertheless, a more
robust approach is needed to ascertain the value of the Poisson bracket on
specific generators. For that, let us consider the two-dimensional vector
space $V=\mathbb{C}x\oplus\mathbb{C}y$ and define the following symplectic
form on it

$\omega(x,y)=-\omega(y,x)=1,\qquad\omega(x,x)=\omega(y,y)=0.$

There is a Leibniz algebra structure [22] on the free noncommutative
$\mathbb{C}$-algebra $R$ generated by two elements $x$ and $y$, defined as
follows:

$\\{u_{1}\cdots u_{p},v_{1}\cdots
v_{q}\\}=\sum\limits_{\begin{subarray}{c}1\leq i\leq p\\\ 1\leq j\leq
q\end{subarray}}\omega(u_{i},v_{j})u_{i+1}\cdots u_{p}u_{1}\cdots
u_{i-1}v_{j+1}\cdots v_{q}v_{1}\cdots v_{j-1},$ (2.5)

where the elements $u_{1},\dots,u_{p},v_{1},\dots,v_{q}$ are either $x$ or
$y$. For the definition and further details regarding Leibniz algebras see
[24]. Now, we define the subspace $\textrm{Com}(R)=\textrm{Span}\\{ab-ba\mid
a,b\in R\\}$ and note that it belongs to the center of $R$. Hence,
$\textrm{Com}(R)$ is a central ideal of $R$ and one can consider the quotient
Leibniz algebra $\mathcal{N}=R/\text{Com}(R)$. Furthermore, this quotient
appears to be a Lie algebra [17]. This Lie algebra is a particular example of
a Necklace Lie algebra, and for a comprehensive combinatorial definition
employing quivers, we refer the reader to [5]. It holds a central position at
the core of our work, as it facilitates the computational aspect through the
application of the Lie algebra structure to the invariants of pairs of
matrices. The trace map defined as follows (see [17])

$\begin{split}tr\colon\mathcal{N}&\rightarrow C_{n2}\\\
x^{k_{1}}y^{l_{1}}\cdots
x^{k_{m}}y^{l_{m}}&\mapsto\big{(}(X,Y)\mapsto\mathrm{Tr}(X^{k_{1}}Y^{l_{1}}\cdots
X^{k_{m}}Y^{l_{m}})\big{)}\end{split}$

is a well-defined Lie algebra homomorphism, which equips us with tools to
compute the brackets on $C_{n2}$, turning the later to a Poisson algebra.
Using this approach, an alternative way to establish the defining relation of
$C_{n2}$ for $n=3$ was offered in [16].

###  2.7. Necklaces

A necklace in $\mathcal{N}$ is formed by equivalence classes of cyclic
permutations of words in a $2$-letter alphabet and, as a standard
consideration, we shall use as representative the biggest one with respect to
the alphabetical order. The usual degree lexicographical order in the space of
two letter words induces a _degree lexicographical order_ in the space of
necklaces by comparing the representative word of their respective equivalence
class.

######  Definition 2.8.

A necklace is said to be $CH_{n}$ if at least one of the words in its
equivalence class contains the same subword $n$ times consecutively.

######  Definition 2.9.

For a fixed bidegree $(r,s)$, we say that a pair of necklaces $(w_{1},w_{2})$
is a _breaking pair_ if it satisfies the following properties:

  * •

The sum of their bidegrees is exactly $(r+1,s+1)$,

  * •

The first one has at least degree $2$ and it is smaller with respect the
degree lexicographical order,

  * •

If the degree of the first necklace is $2$, then the second necklace is
$CH_{n}$,

  * •

The trace of at least one of them is not a term in the linear expansion of the
minimal set of generators.

These pairs will be actively used to compute the value of the algebraic
expressions of the traces of necklaces that are not $CH_{4}$. For instance,
for $n=4$ in bidegree $(3,2)$ the pair $(B^{2},A^{4}B)$ is a breaking pair,
but the pair $(A^{2}B,ABAB)$ is not, since $\operatorname{Tr}(A^{2}B)$ is
exactly $a_{7}$ and $\operatorname{Tr}(ABAB)$ appears in the linear expansion
of $a_{15}$.

##  3\. Description of the algorithmic approach

###  3.1. Overview

Let us sketch the general idea of our procedure. First, we shall inductively
obtain the unique description of the trace values of all necklaces of degree
less or equal than $11$ in terms of the generators from Appendix A. We shall
do this using the Cayley-Hamilton theorem for $CH_{4}$ when it is possible,
and breaking pairs when it is not. With this information we shall obtain the
complete description of the Poisson adjoint maps $\\{a_{5},-\\}$ and
$\\{a_{6},-\\}$ on the generators $a_{3},\dots,a_{32}$.

Then, starting in degree $12$, the approach above will give us the description
of each trace of necklace modulo relations. Whenever a new relation emerges,
it is incorporated to the ideal of relations together with its Poisson bracket
with $a_{5}$ and $a_{6}$. Using the already known Hilbert series, we can stop
the computation when the full dimension is reached and shift to the next step.

In this way, we can obtain almost the whole ideal of relations. The last piece
of the puzzle will be unraveled by a careful exploration of the central
elements of the Necklace Lie algebra.

###  3.2. Initial considerations

The ambivalence between $A$ and $B$ will allow us to work only in bidegrees
$(r,s)$ with $r\geq s$. This inner symmetry will appear in all aspects of our
computations, including the expressions of traces of necklaces, relations and
Poisson brackets.

While finding the value of the bracket of two necklaces in terms of generic
matrices is very fast, when they are replaced by their traceless versions this
computation can be far more costly by the big amount of substitutions that
have to be made. Because of this, it is recommended that anytime a bracket is
computed its value should be stored in the memory, since it will probably be
used again in a higher degree computation.

###  3.3. First non-trivial degree

To demonstrate our approach, we shall commence in degree $n=5$, since up to
degree $4$ all the traces are simply linear combinations of the generators.

We start in bidegree $(5,0)$, where we only find one expression,
$\operatorname{Tr}(A^{5})$, which can be decomposed using the Cayley-Hamilton
theorem and it is equal to $\frac{5}{6}a_{3}a_{6}$. Similarly, in bidegree
$(4,1)$ we get

$\displaystyle\operatorname{Tr}(A^{4}B)$
$\displaystyle=\dfrac{1}{2}\operatorname{Tr}(A^{2})\operatorname{Tr}(A^{2}B)+\dfrac{1}{3}\operatorname{Tr}(A^{3})\operatorname{Tr}(AB)$
$\displaystyle=\dfrac{1}{2}a_{3}a_{7}+\dfrac{1}{3}a_{4}a_{6}.$

In bidegree $(3,2)$ things get more interesting. There are two traces,
$\operatorname{Tr}(A^{3}B^{2})$ and $\operatorname{Tr}(A^{2}BAB)$ that are not
$CH_{4}$. In this bidegree, there is only one breaking pair: $(B^{2},A^{4}B)$.
On one hand, the Poisson bracket of the traces of this pair is:

$\\{\operatorname{Tr}(B^{2}),\operatorname{Tr}(A^{4}B)\\}=-4\operatorname{Tr}(A^{3}B^{2})-4\operatorname{Tr}(A^{2}BAB).$

On the other hand,

$\displaystyle\\{\operatorname{Tr}(B^{2}),\operatorname{Tr}(A^{4}B)\\}$
$\displaystyle=\\{\operatorname{Tr}(B^{2}),\dfrac{1}{2}\operatorname{Tr}(A^{2})\operatorname{Tr}(A^{2}B)+\dfrac{1}{3}\operatorname{Tr}(A^{3})\operatorname{Tr}(AB)\\}$
$\displaystyle=4\operatorname{Tr}(AB)\operatorname{Tr}(AB^{2})+2\operatorname{Tr}(A^{2}B)\operatorname{Tr}(B^{2})+\dfrac{2}{3}\operatorname{Tr}(A^{2})\operatorname{Tr}(B^{3}).$

Equalizing both equations, together with the actual definition of the
generator

$a_{16}=\operatorname{Tr}([A,B]^{2}A)=-\operatorname{Tr}(A^{3}B^{2})+\operatorname{Tr}(A^{2}BAB),$

we obtain that:

$\displaystyle\operatorname{Tr}(A^{3}B^{2})$
$\displaystyle=\frac{1}{12}(a_{5}a_{6}+6a_{4}a_{7}+3a_{3}a_{8}-6a_{16}),$
$\displaystyle\operatorname{Tr}(A^{2}BAB)$
$\displaystyle=\frac{1}{12}(a_{5}a_{6}+6a_{4}a_{7}+3a_{3}a_{8}+6a_{16}).$

This completes the case $n=5$, since the remaining expressions can be
immediately found by making use of the inner symmetry.

###  3.4. Degrees less than 12

It is known that relations appear starting from degree $12$, so
$\operatorname{Tr}(w)$ has a unique expression in terms of the generators for
each $w$ of degree less or equal than $11$. Our task will be to efficiently
compute these expressions. Let us proceed inductively by degree, denoted by
$n$. An immediate application of the Cayley-Hamilton theorem to the traces of
the necklaces that are $CH_{4}$ reduces them to an algebraic expression
involving only traces of lower degree necklaces. Therefore, it will give us
their unique description in terms of the generators.

Then, to find the expressions of the necklaces that are not $CH_{4}$, we shall
proceed by bidegree $(r,s)$ starting with the highest $r$ possible such that
$r+s=n$ where a non-$CH_{4}$ necklace appears.

Let us consider all possible breaking pairs of bidegree $(r,s)$. For each such
pair $(w_{1},w_{2})$, we directly compute the bracket
$\\{\operatorname{Tr}(w_{1}),\operatorname{Tr}(w_{2})\\}$. This results in a
linear combination of traces of necklaces of bidegree $(r,s)$, in which the
traces of $CH_{4}$ necklaces are further substituted by their corresponding
algebraic expression on the generators.

The properties of breaking pairs guarantee us that at least one of the
$\operatorname{Tr}(w_{i})$ can be substituted by an algebraic expression of
lower terms, thus, by the generators. Then, by making these substitution
before computing the bracket
$\\{\operatorname{Tr}(w_{1}),\operatorname{Tr}(w_{2})\\}$, we obtain another
expression just in lower degree terms.

Equalizing both expressions, we obtain a linear equation where the variables
are traces of non-$CH_{4}$ necklaces. Doing this for all breaking pairs
results in a linear system of equations. Furthermore, if there is a generator
$a_{i}$ in this bidegree, we add the equation formed by its expansion as a
linear combination of traces of necklaces. The solution of this completely
determined system will give us the unique algebraic expression of all traces
of non-$CH_{4}$ necklaces in terms of the generators. Then, we continue to
$(r-1,s+1)$ recursively.

Note that to efficiently compute these expressions it is not necessary to
calculate the linear equations for all breaking pairs. We can randomly choose
one pair at a time and add the resulting corresponding equation one by one,
until we reach a completely determined system.

###  3.5. Higher degrees

Starting from degree $12$ we know that relations will start to occur.
Consequently, the expressions of traces of necklaces will stop being unique,
since they are modded by this ideal of relations. Whenever we have to save
these values, it does not matters which representative we choose. We will
inductively construct the ideal $I$ by gathering the relations that will
appear in each step.

The same process described above will now lead to an inconsistent system of
equations. It is an easy linear algebra exercise to find the minimal set of
polynomials on the generators that will be incorporated to $I$ to make the
system consistent. There is a special case to be considered and it is bidegree
$(n/2,n/2)$ for even $n$. In this case, all the breaking pairs are not enough
to generate all the relations, whilst exactly one might be missing. The way to
compute this extra needed equation comes from a careful exploration of the
center of the Necklace Lie algebra. Let us consider the expression
$\operatorname{Tr}([A,B]^{n/2})$. On one hand, we can expand it and substitute
the traces of the $CH_{4}$ necklaces. On the other hand, we can use the
Cayley-Hamilton theorem to break $[A,B]^{n/2}$ into smaller degree terms,
whose expressions on generators are already known. The equalization of both
expansions will be incorporated to the system of equations.

After obtaining all the relations in degree $n$, we compute the Hilbert series
of $I$ and compare it with (2.2). If they do not coincide, we move to the next
degree.

###  3.6. Refined execution

Although the algorithmic approach described above is valid, evidently an
optimization is necessary. Calculating explicit Hilbert series or computing
Poisson brackets at higher degrees can be very demanding. For this reason, we
found some shortcuts using the Poisson structure and general theory of Gröbner
bases.

We still need to compute the expressions of the traces of necklaces, but we
found experimentally that this will only be needed until degree $16$. Let us
explain how to further optimize this process.

Anytime that we update the ideal, we compute a reduced graded Gröbner basis of
it, so the ideal membership problem or the computation of Hilbert series is
fast.

Therefore, after degree $12$, at each bidegree $(r,s)$ we know how many
dimensions we are missing by an examination of the Hilbert series of our
current ideal $I$. If this number is greater than zero, we try to find new
independent elements coming from the Poisson bracket of $a_{5}$ with the
relations of bidegree $(r+1,s-1)$. If they are not enough, we try to find new
polynomials coming from the Poisson bracket of $a_{6}$ with the relations of
bidegree $(r-2,s+1)$.

Now, to compute the expressions of the traces of non-$CH_{4}$ necklaces we can
proceed in the following way. We select randomly one breaking pair at a time
and we consider the resulting equation from the procedure described before.
Depending on its situation with the previously obtained equations we do the
following

  * •

If it is linearly dependent, we discard it.

  * •

If it is linearly independent and does not make the system incompatible, we
add it.

  * •

If it makes the system incompatible, we compute the relation in terms of the
generators, we incorporate it to $I$ if it does not belong to it yet, and then
we discard it.

When the system of equations reaches full rank and $I$ has full dimension, we
stop the procedure. Note that if we are in bidegree $(n/2,n/2)$ for even $n$,
the first equation to consider should be the special one coming from
$\operatorname{Tr}([A,B]^{n/2})$.

When we have all the relations up to degree $16$, we found that we do not need
to compute the expressions of the traces of necklaces any more. The
consecutive application of the Poisson bracket of $a_{5}$ or $a_{6}$ to the
already obtained relations fills up the dimensions of our graded ideal. This
releases us from making a huge amount of computations until $20$, degree where
the last new relation appears.

###  3.7. Outcome

After applying the procedure explained above, we obtained the following data:

The polynomial ideal $I$ is generated by the 105 relations listed in the
ancillary file. The maximum degree found in this set of generators is
$(10,10)$. Furthermore, only 8 relations (also listed in the ancillary file)
are needed to generate $I$ under the repeated adjoint maps $\\{a_{5},-\\}$ and
$\\{a_{6},-\\}$. More precisely, three of them appear in bidegrees:

$(6,6),\quad(7,7),\quad(8,8),$

directly from the expansion of the respective power of the commutator. The
other five are in bidegrees:

$(7,5),\quad(6,6),\quad(7,6),\quad(8,6),\quad(8,7).$

Therefore, we can state the main theorem:

######  Theorem 3.8.

There is an isomorphism

$\mathbb{C}[\mathcal{C}_{42}]\cong\mathbb{C}[a_{1},a_{2}]\otimes\dfrac{\mathbb{C}[a_{3},\dots,a_{32}]}{I}$

###### Proof.

With the help of any computer algebra system it is an easy computation to
check that the 105 polynomials generating $I$ are relations in
$\mathbb{C}[\mathcal{C}_{42}]$. Then, checking that the Hilbert series of $I$
coincides with (2.2) proves the theorem. ∎

###  3.9. Final considerations

The method discussed above is a detailed explanation on how we obtained the
full set of relations generating the ideal. Since the validity of the main
theorem can be checked without examining following the detailed procedure
3.1–3.6, we consider that it is cleaner and clearer to present it in this way.

Since we are working over a polynomial ring of $30$ variables with an ideal
generated by $105$ polynomials—some of them huge—the computation of Gröbner
bases can be a really demanding task even for modern computers. The only way
we were able to calculate the Hilbert series to conclude the proof of Theorem
3.8 was using the Hilbert driven Gröbner basis algorithm implemented in
Macaulay2 [18], with the grading induced by the sum of the lengths. The only
monomial order we could find for this to be executed in a reasonable amount of
time is the following: Let $a_{i},a_{j}$ be of bidegree $(r_{i},s_{i})$ and
$(r_{j},s_{j})$, respectively. Then $a_{i}<a_{j}$ whenever $(r_{i},s_{i})$ is
lower than $(r_{j},s_{j})$ with respect to the lexicographical order. There
are exactly three cases when we have two generators of the same bidegree,
choosing $a_{15}<a_{12}$, $a_{21}<a_{19}$ and $a_{27}<a_{25}$.

The main reason for the high efficiency of this method is the ability to
transition from lower to higher degree relations via the Poisson structure.
Previous approaches were limited to generating relations within the same
degree, and having to again fully decompose the problem when moving to a
higher total grading. Since the highest degree required is 20, the
decomposition should be carried out up to that level. However, with our
method, the computation can stop at degree 16, the highest among our 8
generating relations. In fact, not even the entire degree 16 needs to be
computed, as only one of the extra relations is missing, therefore, a partial
analysis of bidegree (8, 8) suffices.

###  3.10. Hironaka decomposition for $C_{42}$

After getting the full ideal of relations, to get the Hironaka decomposition
of $C_{42}$ becomes a much easier task. After an examination of the modified
denominator (2.3) of the Hilbert series, we choose as primary invariants the
$17$ traces (2.4) denoted by $a_{1},\dots,a_{15},a_{18},a_{20}$ in Appendix A.
Then, we compute a Gröbner basis with respect to the graded reverse
lexicographical order of the ideal generated by
$a_{1},\dots,a_{15},a_{18},a_{20}$ together with the $105$ relations obtained
in Theorem 3.8. Finally, it is a matter of checking which are the 48 normal
monomials in this Gröbner basis.

######  Corollary 3.11.

The algebra $C_{42}$ is a free
$\mathbb{C}[a_{1},\dots,a_{15},a_{18},a_{20}]$-module of rank $48$, and a
basis is listed in Appendix B.

##  4\. Applications to the invariant commuting variety and the Calogero-Moser
space

Since the invariant commuting variety is a subvariety of the variety of pairs
of matrices with respect to conjugation, there is a surjective algebra
homomorphism from $C_{n2}$ to $\mathbb{C}[\operatorname{Com}_{n}]$, for
arbitrary $n>1$. Therefore, we can attempt to leverage the structure of
$C_{n2}$ to obtain the complete description of
$\mathbb{C}[\operatorname{Com}_{n}]$.

If $C_{n2}$ and $\mathbb{C}[\operatorname{Com}_{n}]$ are isomorphic to the
polynomial ideals $\mathbb{C}[a_{1},\dots,a_{k}]/I$ and
$\mathbb{C}[a_{1},\dots,a_{s}]/J$, respectively, where the $a_{i}$ represent
the trace of some expression, there is a well-defined surjection
$\mathbb{C}[a_{1},\dots,a_{k}]\to\mathbb{C}[a_{1},\dots,a_{s}]$ sending any
generator to its corresponding expression in the commuting variety. In other
words, we obtain the following diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}[a_{1},\dots,a_{k}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}[a_{1},\dots,a_{s}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dfrac{\mathbb{C}[a_{1},\dots,a_{k}]}{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\dfrac{\mathbb{C}[a_{1},\dots,a_{s}]}{J}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$

Note that all arrows are algebra homomorphisms, and furthermore, the lower
horizontal arrow is also a Poisson algebra homomorphism.

If we know the description of $I$ and its induced morphism to $J$ happens to
be a surjection, we will obtain the full description of $J$. Let us examine
what happens in the known cases:

For $n=2$, it is established that $k=s=5$, hence the surjection in the middle
is an isomorphism. Nevertheless, $I=0$ but $J=\langle
a_{4}^{2}-a_{3}a_{5}\rangle$, indicating that the induced morphism between the
ideals of relations can never be a surjection.

When $n=3$, it is known that $k=11$ and $s=9$ and the morphism in the middle
sends each generator to itself, except the last two that are mapped to zero.
In this case, $I$ is the ideal generated by a single polynomial, while $J$ is
an ideal generated by five polynomial relations [28]. Therefore, it is obvious
that, again, the induced morphism is not a surjection.

However, for $n=4$ the number of relations increases significantly, so the
situation may differ. It is known that $k=32$ while $s=14$ and, moreover, the
polynomial ideal $I$ is generated by $105$ equations while the ideal $J$ is
generated by just $15$ (see [13]). Looking at the form of the generators in
Appendix A, it is easy to guess the map—$a_{i}$’s are mapped to themselves for
all $1\leq i\leq 14$ and to zero otherwise. Given that the rank of the
Hironaka decomposition of $\mathbb{C}[\operatorname{Com}_{4}]$ is already
established [14], it is a straightforward exercise to verify that this map
induces a surjection from $I$ to $J$, recovering the description of $J$
obtained in [13].

In the case of Calogero-Moser spaces, things work in a similar fashion. We can
construct a diagram as before, with the only variation being the image of the
generators.

For $n=2$, the map is defined identically, but $J=\langle
a_{4}^{2}-a_{3}a_{5}+1\rangle$, so again, it is not possible to recover $J$ by
this map.

When $n=3$, it is known that $s=9$ and the map is defined as follows: the
first nine generators are mapped to themselves while the last two go to the
scalars $-3$ and $2$, respectively [28]. Once more, $I$ is generated by just
one relation, whereas $J$ is generated by five of them. Consequently, $J$
cannot be recovered by this map.

In the case of $n=4$, the situation is analogous to that in the invariant
commuting variety; however, determining the morphism is more intricate.
Motivated by the identities that emerge when the pair of matrices is
restricted to Calogero-Moser space $\mathcal{C}_{4}$, it can be proved by
similar methods of [28] that the generators are mapped as follows:

$\displaystyle a_{i}\mapsto a_{i},\quad\quad 1\leq i\leq 14,$ $\displaystyle
a_{15}\mapsto 6,\quad\quad a_{21}\mapsto 24,\quad a_{27}\mapsto 42,\quad
a_{32}\mapsto 168,$ $\displaystyle a_{18}\mapsto 3a_{3},\quad a_{19}\mapsto
6a_{4},\quad a_{20}\mapsto 3a_{5},$ $\displaystyle a_{24}\mapsto 6a_{3},\quad
a_{25}\mapsto 6a_{4},\quad a_{26}\mapsto 6a_{5},$ $\displaystyle a_{28}\mapsto
6a_{6},\quad a_{29}\mapsto 6a_{7},\quad a_{30}\mapsto 6a_{8},\quad
a_{31}\mapsto 6a_{9}$

and the unmentioned generators are mapped to zero. Analogously, in this case,
the induced morphism from $I$ to $J$ can be easily checked that it is a
surjection, making use of the rank of the Hironaka decomposition of
$\mathbb{C}[\mathcal{C}_{4}]$, recovering the main result of [13].

##  Appendix A Generators

Degree | Generators
---|---
1 | $a_{1}=\operatorname{Tr}(X),\ a_{2}=\operatorname{Tr}(Y)$
2 | $a_{3}=\operatorname{Tr}(A^{2}),\ a_{4}=\operatorname{Tr}(AB),\ a_{5}=\operatorname{Tr}(B^{2})$
3 | $a_{6}=\operatorname{Tr}(A^{3}),\ a_{7}=\operatorname{Tr}(A^{2}B),\ a_{8}=\operatorname{Tr}(AB^{2}),\ a_{9}=\operatorname{Tr}(B^{3})$
4 | $a_{10}=\operatorname{Tr}(A^{4}),\ a_{11}=\operatorname{Tr}(A^{3}B),\ a_{12}=\operatorname{Tr}(A^{2}B^{2}),$
$a_{13}=\operatorname{Tr}(AB^{3}),\ a_{14}=\operatorname{Tr}(B^{4})$
$a_{15}=\frac{1}{2}\operatorname{Tr}([A,B]^{2})$
5 | $a_{16}=\operatorname{Tr}([A,B]^{2}A),\ a_{17}=\operatorname{Tr}([A,B]^{2}B)$
6 | $a_{18}=\operatorname{Tr}([A,B]^{2}A^{2}),$
$a_{19}=\operatorname{Tr}([A,B]^{2}(AB+BA)),$
$a_{20}=\operatorname{Tr}([A,B]^{2}B^{2}),$
$a_{21}=\frac{1}{3}\operatorname{Tr}([A,B]^{3})$
7 | $a_{22}=\operatorname{Tr}([A,B]^{3}A),\ a_{23}=\operatorname{Tr}([A,B]^{3}B)$
8 | $a_{24}=\operatorname{Tr}([A,B]^{3}A^{2}),$
$a_{25}=\frac{1}{2}\operatorname{Tr}([A,B]^{3}(AB+BA)),$
$a_{26}=\operatorname{Tr}([A,B]^{3}B^{2}),$
$a_{27}=\frac{1}{2}\operatorname{Tr}([A,B]^{4})$
9 | $a_{28}=\operatorname{Tr}([A,B]^{3}A^{3}),$
$a_{29}=\frac{1}{3}\operatorname{Tr}([A,B]^{3}(A^{2}B+ABA+BA^{2})),$
$a_{30}=\frac{1}{3}\operatorname{Tr}([A,B]^{3}(AB^{2}+BAB+B^{2}A)),$
$a_{31}=\operatorname{Tr}([A,B]^{3}B^{3}),$
10 | $a_{32}=\operatorname{Tr}([A,B]^{3}(A^{2}B^{2}-AB^{2}A-BA^{2}B+B^{2}A^{2}))$

##  Appendix B Secondary invariants

Degree | Secondary Generators
---|---
0 | 1
5 |  $a_{16}$, $a_{17}$
6 |  $a_{19}$, $a_{21}$
7 |  $a_{22}$, $a_{23}$
8 |  $a_{24}$, $a_{25}$, $a_{26}$, $a_{27}$
9 |  $a_{28}$, $a_{29}$, $a_{30}$, $a_{31}$
10 |  $a_{32}$, $a_{16}^{2}$, $a_{16}a_{17}$, $a_{17}^{2}$
11 |  $a_{16}a_{19}$, $a_{16}a_{21}$, $a_{17}a_{19}$, $a_{17}a_{21}$
12 |  $a_{19}a_{21}$, $a_{21}^{2}$
13 |  $a_{19}a_{22}$, $a_{19}a_{23}$, $a_{21}a_{22}$, $a_{21}a_{23}$
14 |  $a_{19}a_{24}$, $a_{19}a_{26}$, $a_{21}a_{27}$, $a_{22}a_{23}$
15 |  $a_{22}a_{24}$, $a_{23}a_{24}$, $a_{23}a_{25}$, $a_{23}a_{26}$
16 |  $a_{24}a_{25}$, $a_{25}a_{26}$, $a_{25}a_{27}$, $a_{27}^{2}$
17 |  $a_{26}a_{28}$, $a_{26}a_{29}$
18 |  $a_{27}a_{32}$, $a_{29}a_{30}$
19 |  $a_{21}^{2}a_{22}$, $a_{21}^{2}a_{23}$
24 | $a_{25}a_{27}^{2}$

## References

  * [1] S. Abeasis and M. Pittaluga, _On a minimal set of generators for the invariants of $3\times 3$ matrices_, Comm. Algebra 17 (1989), no. 2, 487–499.
  * [2] M. Artin, _On Azumaya algebras and finite dimensional representations of rings_ , J. Algebra 11 (1969), 532–563.
  * [3] H. Aslaksen, V. Drensky, and L. Sadikova, _Defining relations of invariants of two $3\times 3$ matrices_, J. Algebra 298 (2006), no. 1, 41–57.
  * [4] A. Berele and J. R. Stembridge, _Denominators for the Poincaré series of invariants of small matrices_ , Israel J. Math. 114 (1999), 157–175.
  * [5] R. Bocklandt and L. Le Bruyn, _Necklace Lie algebras and noncommutative symplectic geometry_ , Math. Z. 240 (2002), no. 1, 141–167.
  * [6] W. Crawley-Boevey, _Poisson structures on moduli spaces of representations_ , J. Algebra 325 (2011), 205–215.
  * [7] M. Domokos, _Vector invariants of a class of pseudoreflection groups and multisymmetric syzygies_ , J. Lie Theory 19 (2009), no. 3, 507–525.
  * [8] V. Drensky, _Defining relations for the algebra of invariants of $2\times 2$ matrices_, Algebr. Represent. Theory 6 (2003), no. 2, 193–214.
  * [9] V. Drensky and E. Formanek, _Polynomial identity rings_ , Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2004.
  * [10] V. Drensky and R. La Scala, _Defining relations of low degree of invariants of two $4\times 4$ matrices_, Internat. J. Algebra Comput. 19 (2009), no. 1, 107–127.
  * [11] V. Drensky and L. Sadikova, _Generators of invariants of two $4\times 4$ matrices_, C. R. Acad. Bulgare Sci. 59 (2006), no. 5, 477–484.
  * [12] Ya. Dubnov, _Complete system of invariants of two affinors in centro-affine space of two or three dimensions_ , Abh. Sem. Vektor- und Tensoranalysis [Trudy Sem. Vektor. Tenzor. Analizu] 5 (1941), 250–270.
  * [13] F. Eshmatov, X. García-Martínez, Z. Normatov, and R. Turdibaev, _On the coordinate rings of Calogero-Moser spaces and the invariant commuting variety of a pair of matrices_ , arXiv:2307.06098 (2023), 1–15.
  * [14] P. Etingof and V. Ginzburg, _Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism_ , Invent. Math. 147 (2002), no. 2, 243–348.
  * [15] E. Formanek, _The invariants of $n\times n$ matrices_, Invariant theory, Lecture Notes in Math., vol. 1278, Springer, Berlin, 1987, pp. 18–43.
  * [16] X. García-Martínez, Z. Normatov, and R. Turdibaev, _The ring of invariants of pairs of 3 $\times$ 3 matrices_, J. Algebra 603 (2022), 201–212.
  * [17] V. Ginzburg, _Non-commutative symplectic geometry, quiver varieties, and operads_ , Math. Res. Lett. 8 (2001), no. 3, 377–400.
  * [18] D. R. Grayson and M. E. Stillman, _Macaulay2, a software system for research in algebraic geometry_ , Available at http://www.math.uiuc.edu/Macaulay2/.
  * [19] H. Helling, _Eine Kennzeichnung von Charakteren auf Gruppen und assoziativen Algebren_ , Comm. Algebra 1 (1974), 491–501.
  * [20] M. Hochster and J. L. Roberts, _Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay_ , Advances in Math. 13 (1974), 115–175.
  * [21] T. Hoge, _A presentation of the trace algebra of three $3\times 3$ matrices_, J. Algebra 358 (2012), 257–268.
  * [22] M. Kontsevich, _Formal (non)commutative symplectic geometry_ , The Gel’fand Mathematical Seminars, 1990–1992, Birkhäuser Boston, Boston, MA, 1993, pp. 173–187.
  * [23] M. Kontsevich and A. L. Rosenberg, _Noncommutative smooth spaces_ , The Gelfand Mathematical Seminars, 1996–1999, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000, pp. 85–108.
  * [24] J.-L. Loday, _Une version non commutative des algèbres de Lie: les algèbres de Leibniz_ , Enseign. Math. (2) 39 (1993), no. 3-4, 269–293.
  * [25] J. Marsden and A. Weinstein, _Reduction of symplectic manifolds with symmetry_ , Reports on Mathematical Physics 5 (1974), 121–130.
  * [26] D. Mumford, _Geometric invariant theory_ , Springer-Verlag, Berlin-New York, 1965.
  * [27] K. Nakamoto, _The structure of the invariant ring of two matrices of degree 3_ , J. Pure Appl. Algebra 166 (2002), no. 1-2, 125–148.
  * [28] Z. Normatov and R. Turdibaev, _Calogero-Moser spaces and the invariants of two matrices of degree 3_ , to appear in Transform. Groups, 2022, doi:10.1007/s00031-022-09776-y.
  * [29] Z. Normatov and R. Turdibaev, _Poisson algebra structure on the invariants of pairs of matrices of degree three_ , J. Algebra Appl. 22 (2023), no. 3, Paper No. 2350065, 10.
  * [30] D. Ž. Đoković, _Poincaré series of some pure and mixed trace algebras of two generic matrices_ , J. Algebra 309 (2007), no. 2, 654–671.
  * [31] C. Procesi, _The invariant theory of $n\times n$ matrices_, Advances in Math. 19 (1976), no. 3, 306–381.
  * [32] Ju. P. Razmyslov, _Identities with trace in full matrix algebras over a field of characteristic zero._ , Izv. Akad. Nauk SSSR Ser. Mat. (1974), 723–756.
  * [33] K. S. Sibirskiĭ, _Algebraic invariants of a system of matrices._ , Sibirsk. Mat. Z̆. (1968), 152–164.
  * [34] Y. Teranishi, _The ring of invariants of matrices_ , Nagoya Math. J. 104 (1986), 149–161.
  * [35] Francesco Vaccarino, _Linear representations, symmetric products and the commuting scheme_ , J. Algebra 317 (2007), no. 2, 634–641.
  * [36] M. Van den Bergh, _Trace rings of generic matrices are Cohen-Macaulay_ , J. Amer. Math. Soc. 2 (1989), no. 4, 775–799.
  * [37] G. Wilson, _Collisions of Calogero-Moser particles and an adelic Grassmannian_ , Invent. Math. 133 (1998), no. 1, 1–41, With an appendix by I. G. Macdonald.


# Generalized Turán numbers for the edge blow-up of a graph

Zequn Lv  Alfréd Rényi Institute of Mathematics, Hungarian Academy of
Sciences.  Department of Mathematical Sciences, Tsinghua University.  Ervin
Győri  Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences.
Zhen He111Correspondoing author. Email<EMAIL_ADDRESS>Alfréd
Rényi Institute of Mathematics, Hungarian Academy of Sciences.  Department of
Mathematical Sciences, Tsinghua University.  Nika Salia  Alfréd Rényi
Institute of Mathematics, Hungarian Academy of Sciences.  Extremal
Combinatorics and Probability Group, Institute for Basic Science, Daejeon,
South Korea.  Casey Tompkins  Alfréd Rényi Institute of Mathematics, Hungarian
Academy of Sciences.  Kitti Varga  Alfréd Rényi Institute of Mathematics,
Hungarian Academy of Sciences.  Department of Computer Science and Information
Theory, Budapest University of Technology and Economics.  Xiutao Zhu  Alfréd
Rényi Institute of Mathematics, Hungarian Academy of Sciences.  Department of
Mathematics, Nanjing University.

###### Abstract

Let $H$ be a graph and $p$ be an integer. The edge blow-up $H^{p}$ of $H$ is
the graph obtained from replacing each edge in $H$ by a copy of $K_{p}$ where
the new vertices of the cliques are all distinct. Let $C_{k}$ and $P_{k}$
denote the cycle and path of length $k$, respectively. In this paper, we find
sharp upper bounds for ${\rm ex}(n,K_{3},C_{3}^{3})$ and the exact value for
${\rm ex}(n,K_{3},P_{3}^{3})$. Moreover, we determine the graphs attaining
these bounds.

##  1 Introduction

Notation. In this paper, we use $C_{k}$, $P_{k}$, $M_{k}$ and $S_{k}$ to
denote the cycle, path, matching and star with $k$ edges, respectively. Let
$K_{t}$ be the complete graph on $t$ vertices and $K_{s,t}$ be the complete
bipartite graph with parts of size $s$ and $t$. The vertex and edge sets of a
graph $G$ are denoted by $V(G)$ and $E(G)$, respectively. Also we denote the
number of edges in $G$ by $e(G)$. For two graph $G$ and $H$, let $G\cup H$
denote the disjoint union of $G$ and $H$. Let $G+H$ denote the join of $G$ and
$H$, which is obtained from $G\cup H$ by adding all edges with one endvertex
in $V(G)$ and the other endvertex in $V(H)$. Let $T(G)$ denote the set of all
triangles in $G$ and $t(G)=|T(G)|$. For a vertex $v$ in $V(G)$, let $t(v)$
denote the number of triangles containing $v$. For an edge $uv$, let
$N(uv)=N(u)\cap N(v)$. Hence, $|N(uv)|$ is the number of triangles containing
$uv$. For a set of vertices $S\subseteq V(G)$ we denote by $G[S]$ the induced
subgraph of $G$ on $S$ and we set $G-S=G[V(G)-S]$. For two disjoint sets of
vertices $U,W\subseteq V(G)$ we denote by $G[U,W]$ the bipartite subgraph of
$G$ consisting of those edges with one endvertex in $U$ and the other in $W$.

Let $H$ be a given graph and $p$ be an integer greater than $2$. The edge
blow-up $H^{p}$ of $H$ is the graph obtained from replacing each edge in $H$
by a copy of $K_{p}$ where the new vertices of the cliques are all distinct.
The problem of finding the Turán number of $H^{p}$ for various graphs $H$ has
attracted a lot of attention. The first results on the topic can be dated back
to 1960s. Moon [6], and independently Simonovits [7] determined the Turán
number ${\rm ex}(n,M_{k}^{p})$ for $p\geq 3$. Much later Erdős, Füredi, Gould
and Gunderson [2] determined the Turán number ${\rm ex}(n,S_{k}^{p})$ for
$p=3$, and then Chen, Gould, Pfender and Wei [1] extended this result to any
$p\geq 3$. Glebov [4] determined the Turán number of $P_{k}^{p}$. More
recently, Liu extended Glebov’s result to the edge blow-up of a family of
trees and also determined $C_{k}^{p}$ for sufficiently large $n$. Wang, Hou,
Liu and Ma [8] determined the ${\rm ex}(n,T^{P})$ for a larger family of trees
and Yuan [10] determined ${\rm ex}(n,H^{P})$ for any non-bipartite graph $H$
and $p\geq\chi(H)+1$.

We will make use of the following result of Xiao, Katona, Xiao and Zamora [9],
which determined the value of ${\rm ex}(n,C_{3}^{3})$ for all $n\geq 6$.

######  Theorem 1.

(Xiao, Katona, Xiao and Zamora [9]) Let $n\geq 6$ be an integer, then

$\displaystyle{\rm
ex}(n,C_{3}^{3})=\begin{cases}\lfloor\frac{n^{2}}{4}\rfloor+\lfloor\frac{n}{2}\rfloor&\mbox{if
$n\not\equiv 2\ (\mathrm{mod}\ 4)$},\\\
\frac{n^{2}}{4}+\frac{n}{2}-1&\mbox{if\/ $n\equiv 2\ (\mathrm{mod}\ 4)$}.\\\
\end{cases}$

When $n=4k$, $M_{\frac{n}{4}}+M_{\frac{n}{4}}$ is the unique extremal graph.
When $n=4k+1$, $(M_{\lfloor\frac{n}{4}\rfloor}\cup K_{1})+M_{\frac{n-1}{4}}$
and $S_{\lfloor\frac{n}{2}\rfloor}+\overline{K_{\lfloor\frac{n}{2}\rfloor}}$
are the extremal graphs.
When $n=4k+2$, $(M_{\lfloor\frac{n}{4}\rfloor}\cup
K_{1})+(M_{\lfloor\frac{n}{4}\rfloor}\cup K_{1})$,
$M_{\lceil\frac{n}{4}\rceil}+M_{\lfloor\frac{n}{4}\rfloor}$ and
$S_{\frac{n}{2}-1}+\overline{K_{\frac{n}{2}}}$ are the extremal graphs.
When $n=4k+3$, $(M_{\lfloor\frac{n}{4}\rfloor}\cup
K_{1})+M_{\lceil\frac{n}{4}\rceil}$ and
$S_{\lfloor\frac{n}{2}\rfloor}+\overline{K_{\lfloor\frac{n}{2}\rfloor}}$ are
the extremal graphs.

In this paper, we will consider the generalized Turán number. Let $T$ and $H$
be graphs, then the generalized Turán number ${\rm ex}(n,T,H)$ is the maximum
number of copies of $T$ that an $n$-vertex $H$-free graph $G$ can contain. If
$T=K_{2}$, then ${\rm ex}(n,T,H)$ is the classical Turán number of $H$.

Although several results about the Turán number of an edge blow-up of a graph
have been obtained, less is known about the generalized Turán number of such
graphs. However, there have been some results in this direction. Liu and Wang
[5] determined the value of ${\rm ex}(n,K_{p},S_{2}^{p})$ and ${\rm
ex}(n,K_{p},M_{2}^{p})$. Later Gerbner and Patkós [3] determined ${\rm
ex}(n,K_{r},S_{2}^{p})$ and ${\rm ex}(n,K_{r},M_{2}^{p})$ for any $r$, $p$,
and Yuan and Yang [11] determined ${\rm ex}(n,K_{3},M_{2}^{3})$ for all $n$.
Recently, Zhu, Chen, Gerbner, Győri and Hama Kairm [12] determined ${\rm
ex}(n,K_{3},S_{k}^{3})$ for any $k$.

Our results concern the edge blow-ups of cycles and paths. We prove the
following theorems.

######  Theorem 2.

Let $n\geq 22$ be an integer, we have

${\rm ex}(n,K_{3},C_{3}^{3})\leq\frac{n^{2}}{4}-1+\mathbbm{1}_{4|n},$

where $\mathbbm{1}_{4|n}$ is the indicator function for $4|n$. Furthermore,
equality holds when $n$ is even and
$M_{\lceil\frac{n}{4}\rceil}+M_{\lfloor\frac{n}{4}\rfloor}$ is the unique
extremal graph.

######  Theorem 3.

Let $n\geq 300^{3}$ be an integer. We have

${\rm ex}(n,K_{3},P_{3}^{3})=\left\lfloor\frac{(n-1)^{2}}{4}\right\rfloor,$

and the unique extremal graph is
$K_{1}+K_{\left\lfloor{\frac{n-1}{2}}\right\rfloor,\left\lceil{\frac{n-1}{2}}\right\rceil}$.

The rest of the paper is organized as follows. In Section 2, we prove Theorem
2. In Section 3, we prove Theorem 3. In Section 4, we mention some problems
about the general case: ${\rm ex}(n,K_{3},C_{k}^{3})$ and ${\rm
ex}(n,K_{3},P_{k}^{3})$.

##  2 Proof of Theorem 2

One can see that when $n$ is even, the graph
$M_{\lceil\frac{n}{4}\rceil}+M_{\lfloor\frac{n}{4}\rfloor}$ contains
$\frac{n^{2}}{4}-1+\mathbbm{1}_{4|n}$ triangles. So our aim is to show that
${\rm ex}(n,K_{3},C_{3}^{3})\leq\frac{n^{2}}{4}-1+\mathbbm{1}_{4|n}$.

Let $n\geq 22$ be an integer and $G$ be an $n$-vertex $C_{3}^{3}$-free graph
with $t(G)$ being the maximum. We may assume every edge is contained in at
least one triangle, otherwise we delete this edge.

We define the weight of $uv$ by

$w(uv)\coloneqq\frac{1}{|N(uv)|}.$

For a triangle $xyz$, its weight is defined by $w(xyz)=w(xy)+w(xz)+w(yz)$. We
will prove the upper bound by making use of the following claims.

######  Claim 1.

For any triangle $xyz$ in $G$,

$1+\frac{1}{n-2}\leq w(xyz)\leq 3,$

or $w(xy)=w(yz)=w(xz)=\frac{1}{3}$ and there exists another two vertices $u,v$
such that $\\{x,y,z,u,v\\}$ induces a copy of $K_{5}^{-}$ or $K_{5}$.

###### Proof.

Since each edge is contained in at least one triangle, without loss of
generality, we have

$\frac{1}{n-2}\leq w(yz)\leq w(xz)\leq w(xy)\leq 1.$

If $w(xy)=1$, then $w(xyz)\geq 1+\frac{2}{n-2}$ and we are done. Next we may
assume $w(xy)\leq\frac{1}{2}$ and we distinguish two cases based on whether
$w(xy)=\frac{1}{2}$ or $w(xy)\leq\frac{1}{3}$ .

First suppose $w(xy)=\frac{1}{2}$ and let $N(xy)=\\{z,z^{\prime}\\}$. If
$w(xz)=\frac{1}{2}$, then $w(xyz)\geq 1+\frac{1}{n-2}$ and we are done. Thus
we may assume $w(xz)\leq\frac{1}{3}$ and let $y^{\prime}\in
N(xz)-\\{y,z^{\prime}\\}$. If $w(yz)\leq\frac{1}{4}$, then we can find a
vertex $x^{\prime}\in N(yz)-\\{x,y^{\prime},z^{\prime}\\}$ and
$\\{x,y,z,x^{\prime},y^{\prime},z^{\prime}\\}$ contains a copy of $C_{3}^{3}$,
a contradiction. Hence $w(yz)=w(xz)=\frac{1}{3}$ and $w(xyz)=\frac{7}{6}\geq
1+\frac{1}{n-2}$, inequality holds since $n\geq 22$.

Now suppose $w(xy)\leq\frac{1}{3}$. Let $u,v\in N(xy)-\\{z\\}$. If
$w(yz)\leq\frac{1}{4}$, then there is a vertex $x^{\prime}\in
N(yz)-\\{u,v,x\\}$. Also we can find a vertex $y^{\prime}\in
N(xz)-\\{y,x^{\prime}\\}$ and another vertex in $\\{u,v\\}$ not equal to
$y^{\prime}$ (say $u\not=y^{\prime}$). Then
$\\{y^{\prime},x^{\prime},u,x,y,z\\}$ contains a copy of $C_{3}^{3}$, a
contradiction. It follows that $w(xz)=w(yz)=w(xy)=\frac{1}{3}$. Furthermore,
if $N(yz)-\\{x\\}$ or $N(xz)-\\{y\\}$ is not equal to $\\{u,v\\}$, then one
can check that we still can find a copy of $C_{3}^{3}$, a contradiction. Hence
$\\{x,y,z,u,v\\}$ induces a copy of $K_{5}^{-}$ or $K_{5}$. ∎

######  Claim 2.

$t(G)\leq e(G).$

###### Proof.

By Claim 1, we have

$t(G)=\sum_{xyz\in T(G)}1\leq\sum_{xyz\in
T(G)}\left(w(xz)+w(yz)+w(xy)\right)=e(G),$

as required. ∎

######  Claim 3.

For any triangle $xyz$, $w(xyz)\geq 1+\frac{1}{n-2}$, i.e., there is no
$K_{5}^{-}$ in $G$.

###### Proof.

Suppose to the contrary that there is a subgraph $H$ of $G$ isomorphic to
$K_{5}$ or $K_{5}^{-}$ induced on the set
$\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$. If $H$ is isomorphic to $K_{5}^{-}$,
then we may assume without loss of generality $v_{4}v_{5}$ is not an edge.

One can check that for any edge $v_{i}v_{j}$ in $H$, $N(v_{i}v_{j})\subseteq
V(H)$. Otherwise we can find a copy of $C_{3}^{3}$. Let $S=(N(v_{4})\cap
N(v_{5}))-V(H)$ if $H$ is isomorphic to $K_{5}^{-}$ and $S=\emptyset$
otherwise.

If $|S|\leq n-10$, then $e(G-V(H))\leq\frac{(n-5)^{2}}{4}+\frac{n-5}{2}$ by
Theorem 1, and we have

$\displaystyle e(G)\leq$ $\displaystyle~{}e(H)+e(G[V(H),V(G)\setminus
V(H)])+e(G-V(H))$ $\displaystyle\leq$
$\displaystyle~{}10+(n-5)+|S|+\frac{(n-5)^{2}}{4}+\frac{n-5}{2}$
$\displaystyle<$ $\displaystyle~{}\frac{n^{2}}{4}-1.$

By Claim 2, it follows that $G$ is not the extremal graph.

If $|S|>n-10$, then $G[S]$ is $P_{3}$-free, otherwise together with
$v_{4},v_{5}$, we can find a copy of $C_{3}^{3}$. Hence
$e(G-V(H))\leq(n-5-|S|)|S|+|S|+\binom{n-5-|S|}{2}$. When $n\geq 22$, we have

$\displaystyle e(G)\leq~{}$ $\displaystyle
9+(n-5)+|S|+(n-5-|S|)|S|+|S|+\binom{n-5-|S|}{2}$ $\displaystyle\leq~{}$
$\displaystyle 8n-56$ $\displaystyle<~{}$ $\displaystyle\frac{n^{2}}{4}-1.$

Again by Claim 2, we are done. ∎

Let $T_{1}(G)=\\{xyz\in T(G):w(xyz)\geq 1+\frac{2}{n}\\}$ and
$T_{2}(G)=T(G)-T_{1}(G)$. We have the following bound on the average weight of
a triangle in $G$.

######  Claim 4.

The average weight of each triangle in $G$ is at least $1+\frac{2}{n}$.

###### Proof.

If $T_{2}(G)$ is empty, then there is nothing to prove. Hence we may assume
$T_{2}(G)\not=\emptyset$.

Let $xyz$ be a triangle in $T_{2}(G)$ with $w(xy)\geq w(xz)\geq w(yz)$. By
Claim 1 and 3, we have $w(xy)\in\\{1,\frac{1}{2}\\}$. If $w(xy)=1$, then

$w(xyz)\geq 1+\frac{2}{n-2}$

which means $xyz\in T_{1}(G)$, a contradiction. So $w(xy)=\frac{1}{2}$. Let
$N(xy)=\\{z,~{}x^{\prime}\\}$. Suppose
$N(xz)-\\{y,x^{\prime}\\}\not=\emptyset$ and $y^{\prime}\in
N(xz)-\\{y,x^{\prime}\\}$. Then either
$N(yz)-\\{x,~{}x^{\prime},~{}y^{\prime}\\}\not=\emptyset$ and we can find a
copy of $C_{3}^{3}$, or $N(yz)=\\{x,~{}x^{\prime},~{}z^{\prime}\\}$ which
means $\\{x,y,z,x^{\prime},y^{\prime}\\}$ contains a copy of $K_{5}^{-}$, or
$w(yz)\geq\frac{1}{2}$ which means $w(xyz)=\frac{3}{2}\geq 1+\frac{2}{n}$. In
all of these cases, we get a contradiction. Hence,
$N(xz)=\\{y,~{}x^{\prime}\\}$ and $w(xy)=w(xz)=\frac{1}{2}$ and so
$w(yz)\leq\frac{2}{n}$. For the edges $x^{\prime}y,~{}x^{\prime}z$, we deduce
that $w(x^{\prime}y)=w(x^{\prime}z)=\frac{1}{2}$. If not, suppose
$w(x^{\prime}y)<\frac{1}{2}$. Let $u$ be in $N(x^{\prime}y)-\\{x,z\\}$. Since
$|N(yz)|>\frac{n}{2}$, let $v$ be in $N(yz)-\\{x,x^{\prime},u\\}$. Then
$\\{u,v,x^{\prime},x,y,z\\}$ contains a copy of $C_{3}^{3}$, a contradiction.
It follows that the triangle $x^{\prime}yz$ is also in $T_{2}(G)$.

Therefore, for any triangle $xyz$ in $T_{2}(G)$, there is a unique triangle
$x^{\prime}yz$ in $T_{2}(G)$ such that $\\{x,x^{\prime},y,z\\}$ induces a copy
of $K_{4}$ and $w(xy)=w(xz)=w(x^{\prime}y)=w(x^{\prime}z)=\frac{1}{2}$. Hence,
we can partition the set of triangles in $T_{2}(G)$ into pairs
$(xyz,~{}x^{\prime}yz)$. For each such pair, we define a mapping

$\phi(xyz,~{}x^{\prime}yz)=\\{xyz,~{}x^{\prime}yz,~{}xx^{\prime}y,~{}xx^{\prime}z\\}.$

Note that since $w(x^{\prime}z)=\frac{1}{2}$ and $N(x^{\prime}z)=\\{x,y\\}$,
then $N(xx^{\prime})\cap N(yz)=\emptyset$ and
$|N(xx^{\prime})|<n-|N(yz)|\leq\frac{n}{2}$. This means
$xx^{\prime}y,xx^{\prime}z$ are in $T_{1}(G)$. Furthermore, by Claim 3, each
triangle is contained in at most one copy of $K_{4}$, so
$xx^{\prime}y,xx^{\prime}z$ do not belong to any other
$\phi(uvw,~{}u^{\prime}vw)$. Since

$\displaystyle w(xyz)+w(x^{\prime}yz)+w(xx^{\prime}y)+w(xx^{\prime}z)$
$\displaystyle=4+\frac{2}{|N(yz)|}+\frac{2}{N(xx^{\prime})}$
$\displaystyle\geq 4+\frac{8}{|N(xx^{\prime})|+|N(yz)|}\geq 4+\frac{8}{n},$

we can transfer the weight of $xx^{\prime}y,~{}xx^{\prime}z$ to
$xyz,~{}x^{\prime}yz$ and ensure the average weight is at least
$1+\frac{2}{n}$. ∎

Now by the Claim 4 and Theorem 1, we have

$\displaystyle t(G)$ $\displaystyle=\sum_{xyz\in
T(G)}1\leq\frac{n}{n+2}\sum_{xyz\in T(G)}1+\frac{2}{n}$
$\displaystyle\leq\frac{n}{n+2}\sum_{xyz\in
T(G)}\left(w(xz)+w(yz)+w(xy)\right)$
$\displaystyle\leq\frac{n}{n+2}e(G)\leq\frac{n^{2}}{4}-1+\mathbbm{1}_{4|n}.$

Equality holds if and only if $e(G)$ attains the maximum and the average
weight of each triangle is exactly $1+\frac{2}{n}$. Hence, by the
characterization of the extremal graphs for ${\rm ex}(n,C_{3}^{3})$ in Theorem
1, we have $G=M_{\lceil\frac{n}{4}\rceil}+M_{\lfloor\frac{n}{4}\rfloor}$ when
$n$ is even. $\hfill\blacksquare$

##  3 Proof of Theorem 3

Let $t(u,v)$ denote the number of triangles containing $u$ or $v$ or both.
First, we use a technique to reduce the proof of Theorem 3 to the case that
each vertex is contained in many triangles. To this end we use the following
lemma.

######  Lemma 1.

Suppose $G$ is a $P_{3}^{3}$-free graph on at least $300$ vertices. If for any
two different vertices $u,v$, we have $t(u),t(v)\geq\frac{n}{2}-1$ and
$t(u,v)\geq n-2$, then $t(G)\leq\left\lfloor\frac{(n-1)^{2}}{4}\right\rfloor$
and equality holds if and only if
$G=K_{1}+K_{\left\lfloor{\frac{n-1}{2}}\right\rfloor,\left\lceil{\frac{n-1}{2}}\right\rceil}$.

First we will show how to deduce Theorem 3 from Lemma 1, then we will prove
Lemma 1.

###  3.1 Proof of Theorem 3 using Lemma 1.

Let $G$ be a $P_{3}^{3}$-free graph on $n$ vertices with $n\geq 300^{3}$ and
$t(G)\geq\left\lfloor\frac{(n-1)^{2}}{4}\right\rfloor$. We initialize
$G_{n}=G$ and define a process of as follows: for $j<n$, let
$G_{j}=G_{j+1}-v_{1}$ if $t(v_{1})<\frac{j+1}{2}-1$ in $G_{j+1}$, or
$G_{j-1}=G_{j+1}-\\{v_{1},v_{2}\\}$ if $t(v_{1},v_{2})<(j+1)-2$ in $G_{j+1}$.
Suppose the process ends at $G_{\ell}$ and for any two vertices $u,v$ in
$G_{\ell}$, we have $t(u),t(v)\geq\frac{\ell}{2}-1$ and $t(u,v)\geq\ell-2$.
Note that

$\binom{\ell}{3}\geq
t(G_{\ell})\geq\left\lfloor\frac{(\ell-1)^{2}}{4}\right\rfloor+\frac{n-\ell}{2}$

Hence $\ell\geq\sqrt[3]{3n}\geq 300$ and by Lemma 1, $G_{\ell}$ contains a
copy of $P_{3}^{3}$, a contradiction.

That is to say, $G_{n}$ satisfies the conditions in Lemma 1 and we are done.
$\hfill\blacksquare$

###  3.2 Proof of Lemma 1.

Let $G=G_{1}\cup\cdots\cup G_{c}$ be a $P_{3}^{3}$-free graph on $n\geq 300$
vertices, where $G_{i}$ are the connected components of $G$, for $1\leq i\leq
c$. We may assume each edge of $G$ is contained in at least one triangle,
otherwise we delete it and the conditions still hold in the resulting graph.
For any two distinct vertices $u,v$, we have $t(u),t(v)\geq\frac{n}{2}-1$ and
$t(u,v)\geq n-2$. It follows that $v(G_{i})\geq\delta(G)\geq\sqrt{n}$.

As mentioned in the introduction, Yuan and Yang [11] determined ${\rm
ex}(n,K_{3},M_{2}^{3})$ for all $n$.

######  Theorem 4.

(Yuan and Yang [11]) For $n\geq 7$, we have

${\rm
ex}(n,K_{3},M_{2}^{3})=\max\left\\{3n-8,\left\lfloor\frac{(n-1)^{2}}{4}\right\rfloor\right\\}.$

Furthermore, $K_{3}+\overline{K}_{n-3}$ or
$K_{1}+K_{\left\lfloor{\frac{n-1}{2}}\right\rfloor,\left\lceil{\frac{n-1}{2}}\right\rceil}$
is the unique extremal graph.

If no $G_{i}$ contains two vertex-disjoint triangles, then since
$v(G_{i})\geq\sqrt{n}\geq\sqrt{300}$, we have
$t(G_{i})\leq\left\lfloor\frac{(v(G_{i})-1)^{2}}{4}\right\rfloor$ by Theorem 4
and

$t(G)=\sum_{i=1}^{c}t(G_{i})\leq\left\lfloor\frac{(n-1)^{2}}{4}\right\rfloor.$

Equality holds if and only if $G$ is connected and
$G=K_{1}+K_{\left\lfloor{\frac{n-1}{2}}\right\rfloor,\left\lceil{\frac{n-1}{2}}\right\rceil}$.

Therefore, we may assume without loss of generality that $G_{1}$ contains two
vertex-disjoint triangles. We define the distance between two vertex-disjoint
triangles as the minimum length of a path with endvertices in the respective
triangles. Among all vertex-disjoint triangle pairs in $G_{1}$, let
$x_{1}y_{1}z_{1},~{}x_{2}y_{2}z_{2}$ be two vertex disjoint triangles whose
distance is the smallest and let $P=x_{1}\cdots y_{2}$ be a path of minimal
length between them. First suppose the length of $P$ is at least $2$. Let
$x_{1}^{+}$ be the vertex adjacent to $x_{1}$ on the path $P$ and let
$x_{1}x_{1}^{+}w$ be a triangle containing the edge $x_{1}x_{1}^{+}$. Then we
either find a copy of $P_{3}^{3}$ if $w\in\\{x_{2},~{}y_{2},~{}z_{2}\\}$, or
we find another two vertex disjoint triangles whose distance is smaller, and
in both cases we obtain a contradiction. Hence we have that $P=x_{1}y_{2}$ is
a single edge.

Note that $x_{1}y_{2}$ is also contained in a triangle and the third vertex of
this triangle must be in $\\{y_{1},z_{1},x_{2},z_{2}\\}$. Without loss of
generality, say $x_{1}y_{2}z_{2}$ is a triangle. Let $S=(N(y_{2})\cap
N(z_{2}))-\\{x_{1},y_{1},z_{1}\\}$. Obviously, we have that $S$ is nonempty
and independent, since $x_{2}\in S$ and $G$ contains no copy of $P_{3}^{3}$.

Suppose $u$ is a vertex in $N(y_{2})-(S\cup\\{x_{1},y_{1},z_{1},z_{2}\\})$ and
$uy_{2}w$ is a triangle containing the edge $uy_{2}$. If $w$ does not belong
to $\\{x_{1},y_{1},z_{1}\\}$, then
$y_{1}z_{1}x_{1},~{}x_{1}z_{2}y_{2},~{}y_{2}uw$ forms a copy of $P_{3}^{T}$.
If $w\in\\{x_{1},y_{1},z_{1}\\}$, then
$y_{1}z_{1}x_{1},~{}wuy_{2},~{}y_{2}z_{2}x_{2}$ forms a copy of $P_{3}^{T}$.
In both cases we have a contradiction. It follows that $N(y_{2})\subset
S\cup\\{x_{1},y_{1},z_{1},z_{2}\\}$ and analogously $N(z_{2})\subset
S\cup\\{x_{1},y_{1},z_{1},y_{2}\\}$.

Since $\delta(G)\geq\sqrt{n}$, then $|S|\geq 2$. We may assume that for each
edge of the form $y_{2}u$ and $z_{2}u$ with $u\in S$, is only contained in the
triangle $y_{2}uz_{2}$. If not, suppose $y_{2}uw$ is a new triangle distinct
from $y_{2}uz_{2}$. Since $S$ is independent, we have
$w\in\\{x_{1},y_{1},z_{1}\\}$. Let $u^{\prime}\in S-\\{u\\}$, then
$y_{1}z_{1}x_{1},~{}wuy_{2},~{}y_{2}u^{\prime}z_{2}$ forms a copy of
$P_{3}^{3}$, a contradiction. Therefore, if we delete the two vertices
$y_{2},~{}z_{2}$ we destroy at most $|S|+9$ triangles. By the condition
$t(u)+t(v)\geq n-2$, we have $|S|\geq n-11$. We obtain that the total number
of triangles is at most

$(n-11)\binom{11}{2}+\binom{11}{3}<\left\lfloor\frac{(n-1)^{2}}{4}\right\rfloor,$

where the equality holds because $n\geq 300$. It follows that $G$ is not the
extremal graph, and we are done.$\hfill\blacksquare$

##  4 Concluding remarks

In this paper, we studied the generalized Turán number of edge blow-ups of the
graphs $C_{3}^{3}$ and $P_{3}^{3}$. It would be interesting to consider the
general case of $C_{k}^{3}$ and $P_{k}^{3}$. In this section, we pose two
conjectures about the generalized extremal numbers of these graphs.

Let $H(n,p,t)$ denote the graph $K_{t-1}+T_{p}(n-t+1)$, where $T_{p}(n-t+1)$
is the balanced $p$-partite complete graph on $n-t+1$ vertices, i.e., the
Turán graph. Let $H^{+}(n,p,t)$ be the graph obtained from $H(n,p,t)$ by
adding an extra edge in any class of $T_{p}(n-t+1)$.

Based on the our results and the Turán number of ${\rm ex}(n,C_{k}^{3})$ and
${\rm ex}(n,P_{k}^{3})$, we pose the following conjecture for the generalized
Turán number.

######  Conjecture 1.

When $k\geq 4$ and $n$ is sufficiently large,
$H(n,2,\lfloor\frac{k-1}{2}\rfloor+1)$ is the unique extremal graph for both
${\rm ex}(n,K_{3},C_{k}^{3})$ and ${\rm ex}(n,K_{3},P_{k}^{3})$ when $k$ is
odd, and $H^{+}(n,2,\lfloor\frac{k-1}{2}\rfloor+1)$ is the unique extremal
graph when $k$ is even.

##  5 Acknowledgements

The research of the authors Győri and Salia was partially supported by the
National Research, Development and Innovation Office NKFIH, grants K132696 and
SNN-135643. The research of Salia was supported by the Institute for Basic
Science (IBS-R029-C4). The research of Tompkins was supported by NKFIH grant
K135800.

## References

  * [1] Guantao Chen, Ronald J Gould, Florian Pfender, and Bing Wei.  Extremal graphs for intersecting cliques.  Journal of Combinatorial Theory, Series B, 89(2):159–171, 2003\.
  * [2] Paul Erdos, Zoltán Furedi, Ronald J. Gould, and David S. Gunderson.  Extremal graphs for intersecting triangles.  Journal of Combinatorial Theory, Series B, 64(1):89–100, 1995.
  * [3] Dániel Gerbner and Balázs Patkós.  Generalized Turán results for intersecting cliques.  arXiv preprint arXiv:2105.07297, 2021.
  * [4] Roman Glebov.  Extremal graphs for clique-paths.  arXiv preprint arXiv:1111.7029, 2011.
  * [5] Erica L.L. Liu and Jian Wang.  The generalized Turán problem of two intersecting cliques.  arXiv:2101.08004, 2021.
  * [6] John W. Moon.  On independent complete subgraphs in a graph.  Canadian Journal of Mathematics, 20:95–102, 1968.
  * [7] Miklós Simonovits.  A method for solving extremal problems in graph theory, stability problems.  In Theory of Graphs (Proc. Colloq., Tihany, 1966), pages 279–319, 1968.
  * [8] Anyao Wang, Xinmin Hou, Boyuan Liu, and Yue Ma.  The Turán number for the edge blow-up of trees.  Discret. Math., 344:112627, 2021.
  * [9] Chuanqi Xiao, Gyula O.H. Katona, Jimeng Xiao, and Oscar Zamora.  The Turán number of the square of a path.  Discrete Applied Mathematics, 307:1–14, 2022.
  * [10] Long-Tu Yuan.  Extremal graphs for edge blow-up of graphs.  J. Comb. Theory, Ser. B, 152:379–398, 2022.
  * [11] Xiaoli Yuan and Weihua Yang.  On generalized Turán number of two disjoint cliques.  Graphs and Combinatorics, 38(4):Article number:116, 2022.
  * [12] Xiutao Zhu, Yaojun Chen, Daniel Gerbner, Ervin Győri, and Hilal Hama Karim.  Maximum number of triangles in ${F}_{k}$-free graphs.  arXiv:2207.10162, 2022.