arXiv subjects

Parham Salehyan

Publications and source records attributed to Parham Salehyan.

6 recordsLinked to original sources

arXiv2026

On the Bourbaki Degree of Plane Projective Curves

Let $F$ be a reduced plane projective curve defined over an algebraically closed field $k$. The Bourbaki degree of $F$, denoted by $\mathrm{Bour}(F)$, measures its freeness and was introduced in \cite{JNS2024}. It is defined as the degree of $R/I_ε$, where $R = k[x,y,z]$ and $I_ε\subseteq R$ is the Bourbaki ideal associated with a minimal generator $ε$ of the module of first syzygies of the Jacobian ideal $J_F$. In this article, we investigate the local Bourbaki degree and apply it to refine the well-known upper bound for the global Bourbaki degree. We then study curves with prescribed Bourbaki degree through the minimal graded free resolution of $R/J_F$, describing the possible resolution forms and expressing $\mathrm{Bour}(F)$ in terms of the corresponding graded shifts. Finally, we study the realization problem in fixed degree. We prove that for $°F \geq 5$, the du Plessis-Wall bounds impose no numerical obstruction to the realization of the Bourbaki degree. Moreover, we show that every possible Bourbaki degree is realized by reduced plane curves of degrees $5$ and $6$.

math.AG↗

arXiv2026

On Two Algebraic Realizations of Schubert Calculus

Schubert calculus on complex Grassmannians can be played by means of differential operators acting on Schur polynomials or Vertex Operators acting on exterior algebras. In this paper we develop this point of view systematically and complement it with a parallel exterior-algebra formalism, leading to what we call, respectively, the \emph{bosonic} and the \emph{fermionic} Schubert calculus. The two alluded realizations are related by (a finite type version of) the boson--fermion correspondence, thereby providing a unified framework connecting Schubert calculus and integrals on the Grassmannian, symmetric functions, exterior algebras and the representation theory of symmetric groups.

math.RT↗

arXiv2019

Schubert Derivations on the Infinite Wedge Power

The {\em Schubert derivation} is a distinguished Hasse-Schmidt derivation on the exterior algebra of a free abelian group, encoding the formalism of Schubert calculus for all Grassmannians at once. The purpose of this paper is to extend the Schubert derivation to the infinite exterior power of a free ${\mathbb Z}$-module of infinite rank (fermionic Fock space). Classical vertex operators naturally arise from the {\em integration by parts formula}, that also recovers the generating function occurring in the {\em bosonic vertex representation} of the Lie algebra $gl_\infty({\mathbb Z})$, due to Date, Jimbo, Kashiwara and Miwa (DJKM). In the present framework, the DJKM result will be interpreted as a limit case of the following general observation: the singular cohomology of the complex Grassmannian $G(r,n)$ is an irreducible representation of the Lie algebra of $n\times n$ square matrices.}

math.AG↗

arXiv2019

The Cohomology of the Grassmannian is a $gl_n$-module

The integral singular cohomology ring of the Grassmann variety parametrizing $r$-dimensional subspaces in the $n$-dimensional complex vector space is naturally an irreducible representation of the Lie algebra of all the $n\times n$ matrices with integral entries. Using the notion of Schubert derivation, a distinguished Hasse-Schmidt derivation on an exterior algebra, we describe explicitly such a representation, indicating its relationship with the celebrated bosonic vertex representation of the Lie algebra of infinite matrices due to Date, Jimbo, Kashiwara and Miwa.

math.AG↗

arXiv2019

On Plücker Equations Characterizing Grassmann Cones

Polynomial solutions to the KP hierarchy are known to be parametrized by a cone over an infinite-dimensional Grassmann variety. Using the notion of Schubert derivation on a Grassmann algebra, we encode the classical Plücker equations of Grassmannians of r-dimensional subspaces in a formula whose limit for $r\rightarrow\infty$ coincides with the KP hierarchy phrased in terms of vertex operators.

math.AG↗

arXiv2013

Vertex Operators Arising from Linear ODEs

The Heisenberg Oscillator Algebra admits irreducible representations both on the ring $B$ of polynomials in infinitely many indeterminates (the {\em bosonic representation}) and on a graded-by-{\em charge} vector space, the {\em semi-infinite} exterior power of an infinite-dimensional ${\mathbf Q}$-vector space $V$ (the {\em fermionic representation}). Our main observation is that $V$ can be realized as the ${\mathbf Q}$-vector space generated by the solutions to a generic linear ODE of {\em infinite order}. Within this framework, the well known {\em boson-fermion} correspondence for the zero charge fermionic space is a consequence of the formula expressing each solution to a linear ODE as a linear combination of the elements of the universal basis of solutions. In this paper we extend the picture for linear ODEs of finite order. Vertex operators are defined and fully described in this case.

math.AG↗