Cyclotomic Research Articles

Invariants of Weyl Group Action and q-characters of Quantum Affine Algebras

Let W be the Weyl group corresponding to a finite dimensional simple Lie algebra $$\mathfrak {g}$$ of rank $$\ell $$ and let $$m>1$$ be an integer. In Inoue (Lett. Math. Phys. 111(1):32, 2021), by applying cluster mutations, a W-action on $$\mathcal {Y}_m$$ was constructed. Here $$\mathcal {Y}_m$$ is the rational function field on $$cm\ell $$ commuting variables, where $$c \in \{ 1, 2, 3 \}$$ depends on $$\mathfrak {g}$$ . This was motivated by the q-character map $$\chi _q$$ of the category of finite dimensional representations of quantum affine algebra $$U_q(\hat{\mathfrak {g}})$$ . We showed in Inoue (Lett. Math. Phys. 111(1):32, 2021) that when q is a root of unity, $$\textrm{Im} \chi _q$$ is a subring of the W-invariant subfield $$\mathcal {Y}_m^W$$ of $$\mathcal {Y}_m$$ . In this paper, we give more detailed study on $$\mathcal {Y}_m^W$$ ; for each reflection $$r_i \in W$$ associated to the ith simple root, we describe the $$r_i$$ -invariant subfield $$\mathcal {Y}_m^{r_i}$$ of $$\mathcal {Y}_m$$ .

Helson zeta functions for characters with finitely many values

AbstractWe show that the analytic continuations of Helson zeta functions can have essentially arbitrary poles and zeroes in the strip (unconditionally), and in the whole critical strip under Riemann Hypothesis, with the function taking values in cubic roots of unity. If the sets of zeroes and poles are symmetric with respect to the real axis, the same can be achieved with taking values . The proof is constructive.

Text {SL}_{2} tilting modules in the mixed case

Using the non-semisimple Temperley–Lieb calculus, we study the additive and monoidal structure of the category of tilting modules for textrm{SL}_{2} in the mixed case. This simultaneously generalizes the semisimple situation, the case of the complex quantum group at a root of unity, and the algebraic group case in positive characteristic. We describe character formulas and give a presentation of the category of tilting modules as an additive category via a quiver with relations. Turning to the monoidal structure, we describe fusion rules and obtain an explicit recursive description of the appropriate analog of Jones–Wenzl projectors.

Block-circulant complex Hadamard matrices

A new method of obtaining a sequence of isolated complex Hadamard matrices (CHM) for dimensions N ⩾ 7, based on block-circulant structures, is presented. We discuss several analytic examples resulting from a modification of the Sinkhorn algorithm. In particular, we present new isolated matrices of orders 9, 10, and 11, which elements are not roots of unity, and also several new multiparametric families of order 10. We note novel connections between certain eight-dimensional matrices and provide new insights toward the classification of CHM for N ⩾ 7. These contributions can find real applications in quantum information theory and constructions of new families of mutually unbiased bases or unitary error bases.

The Intersection of Traditional Wisdom and Modern Education: Unpacking the Potential of Folk Pedagogy in the Context of India’s National Education Policy 2020

This article explores the critical interface of the National Education Policy (NEP) 2020 and the potential of folk pedagogy in enhancing India’s educational landscape. The NEP 2020, with its emphasis on local, indigenous knowledge systems and a transformative vision for India’s education system, provides an unprecedented opportunity for mainstreaming folk pedagogy into contemporary teaching-learning practices. Approved by the Union Cabinet in 2020 National Education Policy ushered in avant-garde reforms to the education policy of India. Folk pedagogy, a term used to describe traditional modes of education practiced by various indigenous and local communities, carries immense value in terms of cultural preservation, community engagement, and learner-centric methodologies. It promotes experiential learning, critical thinking, and creativity by organically integrating local culture, resources, and traditional wisdom into education. Its salient features and key recommendations exceptionally call for flexibility, multi-disciplinarity, critical thinking and ethical values, multilingualism and the three-language formula. An exceptional emphasis on the power of language and the inclusion of mother/regional under the larger context of Indian ethos has been witnessed for the time. NEP 2020 with such emphasis ushers a pedagogical paradigm shift that seeks to reestablish lost Indian pedagogical heritages and their folk orientation, where education is experiential, diverse, holistic, creative, critical, multidisciplinary and multilingual. By intertwining the principles of NEP 2020 with the rich, culturally diverse practices of folk pedagogy, the article envisages a dynamic, inclusive, and effective education system. It underscores the significant role folk pedagogy can play in achieving NEP’s objectives of creating an equitable and vibrant knowledge society by leveraging local knowledge systems and fostering a deep sense of respect towards the rich cultural heritage of India. The article critically reviews the extent to which NEP 2020 allows for the incorporation of folk pedagogy in its framework, and analyzes how this can be made more effective. It addresses challenges and potential strategies for incorporating folk pedagogy into the mainstream educational practices and curriculum development. The article argues for an extensive dialogue and collaboration among educators, policymakers, and community stakeholders to facilitate this integration, ultimately nurturing a pedagogical system that is grounded in the culture, context, and aspirations of learners. This strategic amalgamation could empower India’s education system to meet the future challenges of the rapidly evolving global knowledge economy, while staying true to its unique cultural roots.

On The Roots of Unity in Several Complex Neutrosophic Rings

On The Roots of Unity in Several Complex Neutrosophic Rings

On the $q$-Analogue of Pólya's Theorem

We answer a question posed by Michael Aissen in 1979 about the $q$-analogue of a classical theorem of George Pólya (1922) on the algebraicity of (generalized) diagonals of bivariate rational power series. In particular, we prove that the answer to Aissen's question, in which he considers $q$ as a variable, is negative in general. Moreover, we show that the answer is positive if and only if $q$ is a root of unity.

Proof of a Conjecture Involving Derangements and Roots of Unity

Let $n>1$ be an odd integer, and let $\zeta$ be a primitive $n$th root of unity in the complex field. Via the Eigenvector-eigenvalue Identity, we show that$$\sum_{\tau\in D(n-1)}\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1+\zeta^{j-\tau(j)}}{1-\zeta^{j-\tau(j)}}=(-1)^{\frac{n-1}{2}}\frac{((n-2)!!)^2}{n},$$where $D(n-1)$ is the set of all derangements of $1,\ldots,n-1$.This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $\delta=0,1$ we determine the value of $\det[x+m_{jk}]_{1\leqslant j,k\leqslant n-1}$ completely, where$$m_{jk}=\begin{cases}(1+\zeta^{j-k})/(1-\zeta^{j-k})&\text{if}\ j\not=k,\\\delta&\text{if}\ j=k.\end{cases}$$

A topological model for the coloured Alexander invariants

Coloured Alexander polynomials form a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group Uq(sl(2)) at roots of unity. This sequence recovers the original Alexander polynomial as the first term. We give a topological model for these invariants, showing that they can be obtained as graded intersection pairings between homology classes in a covering of the configuration space in the punctured disc.

The Feudal and Political System in Pakistan: A Historical Analysis of South Punjab (1969-1990)

The term "feudalism" is often used to describe a social order in which land is held by an elite group that is not subservient to a monarchy or government. In return for their service to the king or his kingdom, his subjects are granted land. In the feudal system, the male head of the family (typically a large joint family) is accountable for the maintenance of the home and the land that has been bequeathed to him. It is true that the social structures of feudalisms in Medieval Europe and Medieval India shared some similarities; however, the feudalism that developed in Pakistan (which was a descendant of Indian feudalism) had its own unique roots and characteristics that were shaped by local history (such as the influence and intent of invaders) and culture. According to land reforms adopted in 1959, each individual's yearly produce from 500 irrigated and 1000 non-irrigated acres was capped at 36,000 index units. This study looks at what happened to feudalism in Pakistan's Southern Punjab between 1969 and 1990, and it offers solutions to the problems that led to its decline.

Genus Field and Extended Genus Field of an Elementary Abelian Extension of Global Fields

In the present work we give the construction of the genus field and the extended genus field of an elementary abelian l-extension of a field of rational functions, where l is a prime number. In the Kummer case, if K is contained in a cyclotomic function field, the construction is given using Leopoldt’s ideas by means of Dirichlet characters. Following the definition of Anglès and Jaulent of extended Hilbert class field, we obtain the extended genus field of an elementary abelian l-extension of a field of rational functions.

Canonical Witt formal scheme extensions and p-torsion groups

We study the nth arithmetic jet space of the p-torsion subgroup attached to a smooth commutative formal group scheme. We show that the nth jet space above fits in the middle of a canonical short exact sequence between a power of the formal scheme of Witt vectors of length n and the p-torsion subgroup we started with. This result generalizes a result of Buium on roots of unity.

Mean square value of $L$-functions at $s=1$ for non-primitive characters, Dedekind sumsand bounds on relative class numbers

Let $d_0$ be a given square-free integer. We give an explicit formula $M_{d_0}(p) =A(d_0)(1+B_{d_0}(p)/p)$ for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the non-primitive odd Dirichlet characters modulo $d_0p$ induced by the odd Dirichlet characters modulo an odd prime $p$. Here $d_0\mapsto A(d_0)$ is an explicit multiplicative arithmetic function and $B_{d_0}(f)$ is a twisted sum over the divisors $d$ of $d_0$ of Dedekind sums $s(h,df)$. To prove that $f\mapsto B_{d_0}(f)$ is $d_0$-periodic, we find a new and closed formula for the Dedekind sums $f\mapsto s(a+bf,c+df)$, for fixed integers $a$, $b$, $c$ and $d$. We deduce explicit upper bounds for relative class numbers of cyclotomic number fields.

Representation of hypergeometric products of higher nesting depths in difference rings

A non-trivial symbolic machinery is presented that can rephrase algorithmically a finite set of nested hypergeometric products in appropriately designed difference rings. As a consequence, one obtains an alternative representation in terms of one single product defined over a root of unity and nested hypergeometric products which are algebraically independent among each other. In particular, one can solve the zero-recognition problem: the input expression of nested hypergeometric products evaluates to zero if and only if the output expression is the zero expression. Combined with available symbolic summation algorithms in the setting of difference rings, one obtains a general machinery that can represent (and simplify) nested sums defined over nested products.

Determination of the modular Jacobian varieties J_1(M,MN) with the Mordell–Weil rank zero

In this paper, we determine all modular Jacobian varieties J_1(M,MN) over the cyclotomic fields {{mathbb {Q}}}(zeta _M) with the Mordell–Weil rank zero assuming the Birch–Swinnerton-Dyer conjecture, following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.

Degeneration of K3 surfaces with non-symplectic automorphisms

We prove that a K3 surface with an automorphism acting on the global 2 -forms by a primitive m -th root of unity, m \neq 1,2,3,4,6 , does not degenerate (assuming the existence of the so-called Kulikov models). A key result used to prove this is the rationality of the actions of automorphisms on the graded quotients of the weight filtration of the l -adic cohomology groups of the surface.

Non-Semisimple TQFT's and BPS q-Series

We propose and in some cases prove a precise relation between 3-manifold invariants associated with quantum groups at roots of unity and at generic q. Both types of invariants are labeled by extra data which plays an important role in the proposed relation. Bridging the two sides - which until recently were developed independently, using very different methods - opens many new avenues. In one direction, it allows to study (and perhaps even to formulate) q-series invariants labeled by spin<sup>c</sup> structures in terms of non-semisimple invariants. In the opposite direction, it offers new insights and perspectives on various elements of non-semisimple TQFT's, bringing the latter into one unifying framework with other invariants of knots and 3-manifolds that recently found realization in quantum field theory and in string theory.

Group Elements Whose Character Values are Roots of Unity

Group Elements Whose Character Values are Roots of Unity

Fusion hierarchies, T-systems and Y-systems for the dilute loop models on a strip

We study the dilute loop models on the geometry of a strip of width N. Two families of boundary conditions are known to satisfy the boundary Yang–Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as its T- and Y-systems, for these four boundary conditions and with a generic crossing parameter λ. For rational and thus a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-N expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small N. The present work complements a previous study (Morin-Duchesne and Pearce 2019 J. Stat. Mech. 1909) that investigated the dilute loop models with periodic boundary conditions.

Lattice cohomology and q-series invariants of 3-manifolds

Abstract In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c {{}^{c}} -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q-series which satisfy some remarkable modularity properties and recover SU ⁢ ( 2 ) {{\rm SU}(2)} quantum invariants of 3-manifolds at roots of unity. In particular, our work gives rise to a 2-variable refinement of the Z ^ {\widehat{Z}} -invariant.