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Kohji Yanagawa

Publications and source records attributed to Kohji Yanagawa.

At least 19 recordsLinked to original sources
arXiv2026

Toward the theory on local cohomologies at the ideals given by simplicial posets

For a simplicial poset $P$, Stanley assigned the face ring $A_P$, which is the quotient of the polynomial ring $S:=K[t_x \mid x \in P \setminus \{\widehat{0} \}]$ by the ideal $I_P$. This is a generalization of Stanley-Reisner rings, but $S$ and $A_P$ are not standard graded in this case, and $I_P$ is not a monomial ideal. To establish the foundation of the theory on local cohomology $H_{I_p}^i(S)$ and its injective resolution, we give an explicit description of the graded injective envelope ${}^*\! E_S(S/\mathfrak{p}_x)$, where $\mathfrak{p}_x$is the prime ideal associated with $x \in P$, and analyze their behavior in the graded dualizing complex.

math.AC↗
arXiv2026

Applications of the finite specializations of the $q$-deformed modular group

Recently, Morier-Genoud and Ovsienko introduced the $q$-deformed modular group. For the construction, they first gave a group $G_q \subset \operatorname{GL}(2, \mathbb{Z}[q^{\pm}])$ and then set $\operatorname{PSL}_q(2,\mathbb{Z}):=G_q/Z(G_q)$. We show that, for $\zeta \in \mathbb{C}^*$, $\operatorname{PSL}_q(2,\mathbb{Z})|_{q=\zeta}$ is finite, if and only if so is $G_q(\zeta):=G_q|_{q=\zeta} \subset \operatorname{GL}(2,\mathbb{C})$, if and only if $\zeta=\zeta_n$ for $n=2,3,4,5$, where $\zeta_n$ is a primitive $n$-th root of unity. Although this result is essentially available in the existing literature, we give a precise proof for further developments. For example, we show that the set of traces of elements in $G_q(\zeta)$ and the set of the special values at $\zeta$ of the normalized Jones polynomials of rational links are finite if and only if $\zeta =\zeta_n$ for $n=2,3,4,5$ and also for $n=6$. We also study the quantized Pythagorean triples.

math.QA↗
arXiv2025

Transposes in the $q$-deformed modular group and their applications to $q$-deformed rational numbers

The (right) $q$-deformed rational numbers was introduced by Morier-Genoud and Ovsienko, and its left variant, whose numerators and denominators are essentially the normalized Jones polynomials of rational links, by Bapat, Becker and Licata. These notions are based on continued fractions and the $q$-deformed modular group $\operatorname{PSL}_q(2,\mathbb{Z})$-actions. In this paper, we introduce the \textit{$q$-transpose} for matrices in $\operatorname{PSL}_q(2,\mathbb{Z})$ to refine the basic perspective of the theory. For example, we present a new proof and a refinement of a theorem of Leclere and Morier-Genoud stating that the trace of $A \in \operatorname{PSL}(2,\mathbb{Z})$ is always palindromic and sign coherent. We also show arithmetic/combinatorial results on left $q$-deformed rationals (e.g., the criterion for their palindromicity). Finally, we discuss the connection to the conjecture of Kantarc{\i} O\u{g}uz on circular fence posets.

math.CO↗
arXiv2024

Arithmetic on $q$-deformed rational numbers

Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\frac{r}s>0$, they constructed coprime polynomials $\mathcal{R}_{\frac{r}s}(q),~ \mathcal{S}_{\frac{r}s}(q) \in {\mathbb Z}[q]$ with $\mathcal{R}_{\frac{r}s}(1)=r,~\mathcal{S}_{\frac{r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \equiv r' \pmod{s}$, then $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$. We show that $rr'{\equiv} -1 \pmod{s}$ implies $\mathcal{S}_{\frac{r}s}(q)=\mathcal{S}_{\frac{r'}s}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r \not \equiv r' \pmod{s}$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if $\mathcal{S}_{\frac{r}s}(\zeta)=0$ for $\zeta=(-1+\sqrt{-3})/2$ (resp. $\zeta=i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links.

math.CO↗
arXiv2023

Gröbner fans of Specht ideals

In this paper, we give the Gröbner fan and the state polytope of a Specht ideal $I_λ$ explicitly. In particular, we show that the state polytope of $I_λ$ for a partition $λ=(λ_1, \ldots, λ_m)$ is always a generalized permutohedron, and it is a (usual) permutohedron if and only if $λ_{i-1}=λ_i>0$ for some $i$.

math.AC↗
arXiv2023

Gröbner bases of radical Li-Li type ideals associated with partitions

For a partition $λ$ of $n$, the _Specht ideal_ $I_λ\subset K[x_1, \ldots, x_n]$ is the ideal generated by all Specht polynomials of shape $λ$. In their unpublished manuscript, Haiman and Woo showed that $I_λ$ is a radical ideal, and gave its universal Gröbner bases (recently, Murai et al. published a quick proof of this result). On the other hand, an old paper of Li and Li studied analogous ideals, while their ideals are not always radical. In the present paper, we introduce a class of ideals which generalizes both Specht ideals and _radical_ Li-Li ideals, and study their radicalness and Gröbner bases.

math.AC↗
arXiv2023

Elementary construction of the minimal free resolution of the Specht ideal of shape $(n-d,d)$

Let $K$ be a field with ${\rm char}(K)=0$. For a partition $λ$ of $n \in {\mathbb N}$, let $I^{\rm Sp}_λ$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $λ$. These ideals have been studied from several points of view (and under several names). Using advanced tools of the representation theory, Berkesch Zamaere et al [BGS]. constructed a minimal free resolution of $I^{\rm Sp}_{(n-d,d)}$ except differential maps. The present paper constructs the differential maps, and also gives an elementary proof of the result of [BGS].

math.AC↗
arXiv2021

A note on the reducedness and Gröbner bases of Specht ideals

The Specht ideal of shape $λ$, where $λ$ is a partition, is the ideal generated by all Specht polynomials of shape $λ$. Haiman and Woo proved that these ideals are reduced and found their universal Gröbner bases. In this short note, we give a short proof for these results.

math.AC↗
arXiv2021

Elementary construction of minimal free resolutions of the Specht ideals of shapes $(n-2,2)$ and $(d,d,1)$

For a partition $λ$ of $n \in \mathbb{N}$, let $I^{\rm Sp}_λ$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $λ$. We assume that ${\rm char}(K)=0$. Then $R/I^{\rm Sp}_{(n-2,2)}$ is Gorenstein, and $R/I^{\rm Sp}_{(d,d,1)}$ is a Cohen-Macaulay ring with a linear free resolution. In this paper, we construct minimal free resolutions of these rings. Berkesch Zamaere, Griffeth, and Sam had already studied minimal free resolutions of $R/I^{\rm Sp}_{(n-d,d)}$, which are also Cohen-Macaulay, using heighly advanced technique of the representation theory. However we only use the basic theory of Specht modules, and explicitly describe the differential maps.

math.AC↗
arXiv2021

Regularity of Cohen-Macaulay Specht ideals

For a partition $λ$ of $n \in {\mathbb N}$, let $I^{\rm Sp}_λ$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $λ$. In the previous paper, the second author showed that if $R/I^{\rm Sp}_λ$ is Cohen-Macaulay, then $λ$ is either $(n-d,1,\ldots,1),(n-d,d)$, or $(d,d,1)$, and the converse is true if ${\rm char}(K)=0$. In this paper, we compute the Hilbert series of $R/I^{\rm Sp}_λ$ for $λ=(n-d,d)$ or $(d,d,1)$. Hence, we get the Castelnuovo-Mumford regularity of $R/I^{\rm Sp}_λ$, when it is Cohen-Macaulay. In particular, $I^{\rm Sp}_{(d,d,1)}$ has a $(d+2)$-linear resolution in the Cohen-Macaulay case.

math.AC↗
arXiv2019

Alexander duality for the alternative polarizations of strongly stable ideals

We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal $I \subset \Bbbk[x_1, \ldots, x_n]$ with ${\rm deg}(\mathsf{m}) \le d$ for all $\mathsf{m} \in G(I)$, its dual $I^* \subset \Bbbk[y_1, \ldots, y_d]$ is a strongly stable ideal with ${\rm deg}(\mathsf{m}) \le n$ for all $\mathsf{m} \in G(I^*)$. This duality has been constructed by Fl$ø$ystad et al. in a different manner, so we emphasis applications here. For example, we will describe the Hilbert serieses of the local cohomologies $H_\mathfrak{m}^i(S/I)$ using the irreducible decomposition of $I$ (through the Betti numbers of $I^*$).

math.AC↗
arXiv2019

When is a Specht ideal Cohen-Macaulay?

For a partition $λ$ of $n$, let $I^{\rm Sp}_λ$ be the ideal of $R=K[x_1, \ldots, x_n]$ generated by all Specht polynomials of shape $λ$. We show that if $R/I^{\rm Sp}_λ$ is Cohen--Macaulay then $λ$ is of the form either $(a, 1, \ldots, 1)$, $(a,b)$, or $(a,a,1)$. We also prove that the converse is true if ${\rm char}(K)=0$. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R/I^{\rm Sp}_{(n-3,3)}$ is NOT Cohen--Macaulay if and only if ${\rm char}(K)=2$.

math.AC↗
arXiv2018

Vandermonde determinantal ideals

We show that the ideal generated by maximal minors (i.e., $(k+1)$-minors) of a $(k+1) \times n$ Vandermonde matrix is radical and Cohen-Macaulay. Note that this ideal is generated by all Specht polynomials with shape $(n-k,1,...,1)$.

math.AC↗
arXiv2017

The Cohen-Macaulayness of the bounded complex of an affine oriented matroid

An affine oriented matroid is a combinatorial abstraction of an affine hyperplane arrangement. From it, Novik, Postnikov and Sturmfels constructed a squarefree monomial ideal in a polynomial ring, called an oriented matroid ideal, and got beautiful results. Developing their theory, we will show the following. (1) If an oriented matroid ideal is Cohen-Macaulay, then the bounded complex (a regular CW complex associated with it) of the corresponding affine oriented matroid is a contractible homology manifold with boundary. This is closely related to Dong's theorem, which used to be "Zaslavsky's conjecture". (2) We characterize the affine oriented matroid whose corresponding ideal is Cohen-Macaulay. (3) In the Cohen-Macaulay case, we give a description of the canonical module of the residue class ring by an oriented matroid ideal.

math.AC↗
arXiv2016

Non-level semi-standard graded Cohen-Macaulay domain with $h$-vector $(h_0,h_1,h_2)$

Let $k$ be an algebraically closed field of characteristic 0, and $A$ a Cohen-Macaulay graded domain with $A_0=k$. If $A$ is semi-standard graded (i.e., $A$ is finitely generated as a $k[A_1]$-module), it has the $h$-vector $(h_0, h_1, ..., h_s)$, which encodes the Hilbert function of $A$. From now on, assume that $s=2$. It is known that if $A$ is standard graded (i.e., $A=k[A_1]$), then $A$ is level. We will show that, in the semi-standard case, if $A$ is not level, then $h_1+1$ divides $h_2$. Conversely, for any positive integers $h$ and $n$, there is a non-level $A$ with the $h$-vector $(1, h, (h+1)n)$. Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings).

math.AC↗
arXiv2016

Lyubeznik numbers of local rings and linear strands of graded ideals

In this work we introduce a new set of invariants associated to the linear strands of a minimal free resolution of a $\mathbb{Z}$-graded ideal $I\subseteq R=\Bbbk[x_1, \ldots, x_n]$. We also prove that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. In particular, they satisfy a consecutiveness property that we prove first for Lyubeznik numbers. For the case of squarefree monomial ideals we get more insight on the relation between Lyubeznik numbers and the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik numbers of Stanley-Reisner rings are not only an algebraic invariant but also a topological invariant, meaning that they depend on the homeomorphic class of the geometric realization of the associated simplicial complex and the characteristic of the base field.

math.AC↗
arXiv2016

A Canonical module characterization of Serre's $(\mathrm{R}_1)$

In this short note, we give a characterization of domains satisfying Serre's condition $(\mathrm{R}_1)$ in terms of their canonical modules. In the special case of toric rings, this generalizes a result of the second author (K. Yanagawa, Dualizing complexes of seminormal affine semigroup rings and toric face rings, J. Algebra 425 (2015).) where the normality is described in terms of the "shape" of the canonical module.

math.AC↗