Corrigendum to “Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras” [J. Pure Appl. Algebra 227(4) (2023) 107281]
We define a unified trace form $τ$ on the cyclotomic Hecke algebras $\mathscr{H}_{n,K}$ of type $A$, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter $ξ\neq …
We define a unified trace form $τ$ on the cyclotomic Hecke algebras $\mathscr{H}_{n,K}$ of type $A$, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter $ξ\neq 1$) and Brundan-Kleshchev's trace form on the degenerate version. We use seminormal basis theory to construct a pair of dual bases for $\mathscr{H}_{n,K}$ with respect to the form. We also construct an explicit basis for the cocenter (i.e., the $0$th Hochschild homology) of the corresponding cyclotomic Schur algebra, which shows that the cocenter has dimension independent of the ground field $K$, the Hecke parameter $ξ$ and the cyclotomic parameters $Q_1,\cdots,Q_\ell$.
Let $\mathscr{R}_\alpha^\Lambda$ be the cyclotomic KLR algebra associated to a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$ and polynomials $\{Q_{ij}(u,v)\}_{i,j\in I}$. Shan, Varagnolo and Vasserot show that, when the ground field $K$ …
Let $\mathscr{R}_\alpha^\Lambda$ be the cyclotomic KLR algebra associated to a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$ and polynomials $\{Q_{ij}(u,v)\}_{i,j\in I}$. Shan, Varagnolo and Vasserot show that, when the ground field $K$ has characteristic $0$, the degree $d$ component of the cocenter $Tr(\mathscr{R}_\alpha^\Lambda)$ is nonzero only if $0\leq d\leq d_{\Lambda,\alpha}$. In this paper we show that this holds true for arbitrary ground field $K$, arbitrary $\mathfrak{g}$ and arbitrary polynomials $\{Q_{ij}(u,v)\}_{i,j\in I}$. We generalize our earlier results on the $K$-linear generators of $Tr(\mathscr{R}_\alpha^\Lambda), Tr(\mathscr{R}_\alpha^\Lambda)_0, Tr(\mathscr{R}_\alpha^\Lambda)_{d_{\Lambda,\alpha}}$ to arbitrary ground field $K$. Moreover, we show that the dimension of the degree $0$ component $Tr(\mathscr{R}_\alpha^\Lambda)_0$ is always equal to $\dim V(\Lambda)_{\Lambda-\alpha}$, where $V(\Lambda)$ is the integrable highest weight $U(\mathfrak{g})$-module with highest weight $\Lambda$, and we obtain a basis for $Tr(\mathscr{R}_\alpha^\Lambda)_0$.
In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We …
In this work, we present a explicit trace forms for maximal real subfields of cyclotomic fields as tools for constructing algebraic lattices in Euclidean space with optimal center density. We also obtain a closed formula for the Gram matrix of algebraic lattices obtained from these subfields. The obtained lattices are rotated versions of the lattices $ \Lambda_9, \Lambda_{10}$ and $\Lambda_{11}$ and they are images of $\mathbb{Z}$-submodules of rings of integers under the twisted homomorphism, and these constructions, as algebraic lattices, are new in the literature. We also obtain algebraic lattices in odd dimensions up to $7$ over real subfields, calculate their minimum product distance and compare with those known in literatura, since lattices constructed over real subfields have full diversity.
The cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for …
The cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We here prove an analogue of Jantzen’s sum formula for the cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.
This chapter contains sections titled: Cyclotomic Polynomials Gauss and Roots of Unity (Optional)
This chapter contains sections titled: Cyclotomic Polynomials Gauss and Roots of Unity (Optional)
Let m ≥ 3 be an integer, ζ m ∈ ℂ a primitive mth root of unity, and K m the cyclotomic field ℚ(ζ m ). An explicit description of …
Let m ≥ 3 be an integer, ζ m ∈ ℂ a primitive mth root of unity, and K m the cyclotomic field ℚ(ζ m ). An explicit description of the integral trace form [Formula: see text] where [Formula: see text] is the complex conjugate of x is presented. In the case where m is prime, a procedure for finding the minimum of the form subject to x being a nonzero element of a certain ℤ-module in ℤ[ζ m ] is presented.
The paper uses the cellular basis of the (semi-simple) degenerate cyclotomic Hecke algebras to investigate these algebras exhaustively. As a consequence, we describe explicitly the "Young's seminormal form" and a …
The paper uses the cellular basis of the (semi-simple) degenerate cyclotomic Hecke algebras to investigate these algebras exhaustively. As a consequence, we describe explicitly the "Young's seminormal form" and a orthogonal bases for Specht modules and determine explicitly the closed formula for the natural bilinear form on Specht modules and Schur elements for the degenerate cyclotomic Hekce algebras.
We generalise the study of cyclotomic matrices those with all eigenvalues in the interval [−2, 2] from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers …
We generalise the study of cyclotomic matrices those with all eigenvalues in the interval [−2, 2] from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers of imaginary quadratic fields. As in the rational integer case, a corresponding graph-like structure is defined. We introduce the notion of ‘4-cyclotomic’ matrices and graphs, prove that they are necessarily maximal cyclotomic, and classify all such objects up to equivalence. Six rings OQ(d) for d = −1,−2,−3,−7,−11,−15 give rise to examples not found in the rational-integer case; in four (d = −1,−2,−3,−7) we recover infinite families as well as sporadic cases. For d = −15,−11,−7,−2, we demonstrate that a maximal cyclotomic graph is necessarily 4cyclotomic and thus the presented classification determines all cyclotomic matrices/graphs for those fields. For the same values of d we then identify the minimal noncyclotomic graphs and determine their Mahler measures; no such graph has Mahler measure less than 1.35 unless it admits a rational-integer representative.
Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique multiplicative natural transformation from algebraic K-theory to topological Hochschild homology (THH) and …
Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique multiplicative natural transformation from algebraic K-theory to topological Hochschild homology (THH) and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology (TC). Moreover, we prove that the space of all multiplicative structures on algebraic K-theory is contractible. We also show that the algebraic K-theory functor from small stable infinity categories to spectra is lax symmetric monoidal, which in particular implies that E_n ring spectra give rise to E_{n-1} ring algebraic K-theory spectra. Along the way, we develop a "multiplicative Morita theory", establishing a symmetric monoidal equivalence between the infinity category of small idempotent-complete stable infinity categories and the Morita localization of the infinity category of spectral categories.
Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique natural transformation from algebraic K-theory to topological Hochschild homology (THH) and the …
Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique natural transformation from algebraic K-theory to topological Hochschild homology (THH) and the cyclotomic trace map as the unique lift through topological cyclic homology (TC). Moreover, we prove that the space of all structures on algebraic K-theory is contractible.
We also show that the algebraic K-theory functor from small stable infinity categories to spectra is lax symmetric monoidal, which in particular implies that E_n ring spectra give rise to E_{n-1} ring algebraic K-theory spectra. Along the way, we develop a multiplicative Morita theory, establishing a symmetric monoidal equivalence between the infinity category of small idempotent-complete stable infinity categories and the Morita localization of the infinity category of spectral categories.
The center conjecture for the cyclotomic KLR algebras $R_\beta^\Lambda$ asserts that the center of $R_\beta^\Lambda$ consists of symmetric elements in its KLR $x$ and $e(\nu)$ generators. In this paper we …
The center conjecture for the cyclotomic KLR algebras $R_\beta^\Lambda$ asserts that the center of $R_\beta^\Lambda$ consists of symmetric elements in its KLR $x$ and $e(\nu)$ generators. In this paper we show that this conjecture is equivalent to the injectivity of some natural map $\bar{\iota}_\beta^{\Lambda,i}$ from the cocenter of $R_\beta^\Lambda$ to the cocenter of $R_\beta^{\Lambda+\Lambda_i}$ for all $i\in I$ and $\Lambda\in P^+$. We prove that the map $\bar{\iota}_\beta^{\Lambda,i}$ is given by multiplication with a center element $z(i,\beta)\in R_\beta^{\Lambda+\Lambda_i}$ and we explicitly calculate the element $z(i,\beta)$ in terms of the KLR $x$ and $e(\nu)$ generators. We present an explicit monomial basis for certain bi-weight spaces of the defining ideal of $R_\beta^\Lambda$ and of $R_\beta^\Lambda$. For $\beta=\sum_{j=1}^n\alpha_{i_j}$ with $\alpha_{i_1},\cdots, \alpha_{i_n}$ pairwise distinct, we construct an explicit monomial basis of $R_\beta^\Lambda$, prove the map $\bar{\iota}_\beta^{\Lambda,i}$ is injective and thus verify the center conjecture for these $R_\beta^\Lambda$.
Abstract The center conjecture for the cyclotomic KLR algebras $\mathscr{R}_{\beta }^{\Lambda }$ asserts that the center of $\mathscr{R}_{\beta }^{\Lambda }$ consists of symmetric elements in its KLR $x$ and $e(\nu …
Abstract The center conjecture for the cyclotomic KLR algebras $\mathscr{R}_{\beta }^{\Lambda }$ asserts that the center of $\mathscr{R}_{\beta }^{\Lambda }$ consists of symmetric elements in its KLR $x$ and $e(\nu )$ generators. In this paper, we show that this conjecture is equivalent to the injectivity of some natural map $\overline{\iota }_{\beta }^{\Lambda ,i}$ from the cocenter of $\mathscr{R}_{\beta }^{\Lambda }$ to the cocenter of $\mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$ for all $i\in I$ and $\Lambda \in P^{+}$. We prove that the map $\overline{\iota }_{\beta }^{\Lambda ,i}$ is given by multiplication with a center element $z(i,\beta )\in \mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$ and we explicitly calculate the element $z(i,\beta )$ in terms of the KLR $x$ and $e(\nu )$ generators. We present explicit monomial bases for certain bi-weight spaces of the defining ideal of $\mathscr{R}_{\beta }^{\Lambda }$. For $\beta =\sum _{j=1}^{n}\alpha _{i_{j}}$ with $\alpha _{i_{1}},\cdots , \alpha _{i_{n}}$ pairwise distinct, we construct an explicit monomial basis of $\mathscr{R}_{\beta }^{\Lambda }$, prove the map $\overline{\iota }_{\beta }^{\Lambda ,i}$ is injective, and thus verify the center conjecture for these $\mathscr{R}_{\beta }^{\Lambda }$.
We provide a new construction of the topological cyclic homology $TC(C)$ of any spectrally-enriched $\infty$-category $C$, which affords a precise algebro-geometric interpretation of the cyclotomic trace map $K(X) \to TC(X)$ …
We provide a new construction of the topological cyclic homology $TC(C)$ of any spectrally-enriched $\infty$-category $C$, which affords a precise algebro-geometric interpretation of the cyclotomic trace map $K(X) \to TC(X)$ from algebraic K-theory to topological cyclic homology for any scheme $X$. This construction rests on a new identification of the cyclotomic structure on $THH(C)$, which we find to be a consequence of (i) the geometry of 1-manifolds, and (ii) linearization (in the sense of Goodwillie calculus). Our construction of the cyclotomic trace likewise arises from the linearization of more primitive data.
This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on …
This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on each coset of the index $d$ subgroup of $\mathbb{F}_q^{\ast}$. We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index $d$ generalized cyclotomic permutations of $\mathbb{F}_q$ and pertain to cycle structures, the classification of $(q-1)$-cycles and involutions, as well as inversion.
This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on …
This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree with a suitable monomial function $x\mapsto ax^r$ on each coset of the index $d$ subgroup of $\mathbb{F}_q^{\ast}$. We discuss two important rewriting procedures in the context of generalized cyclotomic mappings and present applications thereof that concern index $d$ generalized cyclotomic permutations of $\mathbb{F}_q$ and pertain to cycle structures, the classification of $(q-1)$-cycles and involutions, as well as inversion.
We show that cyclotomic Sergeev algebra $\mathfrak{h}_n^g$ is symmetric when the level is odd and supersymmetric when the level is even. We give an integral basis for ${\rm Tr}(\mathfrak{h}_n^g)_{\overline{0}}$, and …
We show that cyclotomic Sergeev algebra $\mathfrak{h}_n^g$ is symmetric when the level is odd and supersymmetric when the level is even. We give an integral basis for ${\rm Tr}(\mathfrak{h}_n^g)_{\overline{0}}$, and recover Ruff's result on the rank of ${\rm Z}(\mathfrak{h}_n^g)_{\bar{0}}$ when the level is odd. We obtain a generating set of ${\rm SupTr}(\mathfrak{h}_n^g)_{\overline{0}}$, which gives an upper bound of the dimension of ${\rm Z}(\mathfrak{h}_n^g)_{\bar{0}}$ when the level is even.