pinecone/core-2020-05-10-deduplication huggingface dataset:
Flag Varieties and Interpretations of Young Tableau Algorithms

AbstractThe conjugacy class of nilpotent n×n matrices can be parameterized by partitions λ of n, and for a nilpotent η in the class parameterized by λ, the variety Fη of η-stable flags has its irreducible components parameterized by the standard Young tableaux of shape λ. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Steinberg's result that the relative position of flags generically chosen in the irreducible components of Fη parameterized by tableaux P and Q is the permutation associated to (P,Q) under the Robinson–Schensted correspondence. Other constructions for which we give interpretations are Schützenberger's involution of the set of Young tableaux, jeu de taquin (leading also to an interpretation of Littlewood–Richardson coefficients), and the transpose Robinson–Schensted correspondence (defined using column insertion). In each case we use a doubly indexed family of partitions, defined in terms of the flag (or pair of flags) determined by a point chosen in the variety under consideration, and we show that for generic choices, the family satisfies combinatorial relations that make it correspond to an instance of the algorithmic operation being interpreted (as described in M. A. A. van Leeuwen, Electron. J. Combin.3, No. 2 (1996), R15)
Relative Galois Module Structure and Steinitz Classes of Dihedral Extensions of Degree 8

AbstractLet k be a number field and Ok its ring of integers. Let Γ be the dihedral group of order 8. Let M be a maximal Ok-order in k[Γ] containing Ok[Γ] and Cℓ(M) its class group. We denote by R(M) the set of realizable classes, that is, the set of classes c∈Cℓ(M) such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ and the class of M⊗Ok[Γ]ON equal to c, where ON is the ring of integers of N. In this article we prove that R(M) is a subgroup of Cℓ(M) provided that k and the fourth cyclotomic field of Q are linearly disjoint, and the class number of k is odd. To this end we will solve an embedding problem connected with Steinitz classes of Galois extensions. In addition, for any k with odd class number, we show that the set of Steinitz classes of tame dihedral extension of k is the full class group of k
The Ideals of Free Differential Algebras

AbstractWe consider the free C-algebra Bq with N generators {ξi}i=1,…,N, together with a set of N differential operators {∂i}i=1,…,N that act as twisted derivations on Bq according to the rule ∂iξj=δij+qijξj∂i; that is, ∀x∈B,∂i(ξjx)=δijx+qijξj∂ix, and ∂iC=0. The subscript q on Bq stands for {qij}i,j∈{1,…,N} and is interpreted as a point in parameter space, q={qij}∈CN2. A constant C∈Bq is a nontrivial element with the property ∂iC=0, i=1,…,N. To each point in parameter space there corresponds a unique set of constants and a differential complex. There are no constants when the parameters qij are in general position. We obtain some precise results concerning the algebraic surfaces in parameter space on which constants exist. Let Iq denote the ideal generated by the constants. We relate the quotient algebras B′q=Bq/Iq to Yang–Baxter algebras and, in particular, to quantized Kac–Moody algebras. The differential complex is a generalization of that of a quantized Kac–Moody algebra described in terms of Serre generators. Integrability conditions for q-differential equations are related to Hochschild cohomology. It is shown that Hp(B′q,B′q)=0 for p≥1. The intimate relationship to generalized, quantized Kac–Moody algebras suggests an approach to the problem of classification of these algebras
On Some Classes of Artinian Rings

AbstractA module M is called a CS-module if every submodule of M is essential in a direct summand of M. A ring R is called CS-semisimple if every right R-module is CS. For a ring R, we show that:&#x02022;is right artinian with Jacobson radical cube zero if every countably generated right -module is a direct sum of a projective module and a CS-module.&#x02022;The following conditions are equivalent: (i) Every countably generated right -module is a direct sum of a projective module and a quasicontinuous module; and (ii) every right -module is a direct sum of a projective module and a quasi-injective module.We describe the structure of rings in (2) and show that such a ring is not necessarily CS-semisimple
On Vanishing Sums of Roots of Unity

AbstractAn unsolved problem in number theory asked the following: For a given natural number m, what are the possible integers n for which there exist mth roots of unity α1,…,αn∈C such that α1+···+αn=0? We show in this paper that the set of all possible n's is exactly the collection of N-combinations of the prime divisors of m, where N denotes the set of all non-negative integers. The proof is long and involves a subtle analysis of minimal vanishing sums of mth roots of unity, couched in the setting of integral group rings of finite cyclic groups. Our techniques also recovered with ease some of the classical results on vanishing sums of roots of unity, such as those of Rédei, de Bruijn, and Schoenberg
K0 of a semilocal ring

Let R be a semilocal ring, that is, R modulo its Jacobson radical J(R) is artinian. Then K0(R/J(R)) is a partially ordered abelian group with order-unit, isomorphic to (Zn, ≤, u), where ≤ denotes the componentwise order on Zn and u is an order-unit in (Zn, ≤). Moreover, the canonical projection π:R → R/J(R) induces an embedding of partially ordered abelian groups with order-unit K0(π): K0(R) → K0(R/J(R)). In this paper we prove that every embedding of partially ordered abelian groups with order-unit G → Zn can be realized as the mapping K0(π): K0(R) → K0(R/J(R)) for a suitable hereditary semilocal ring R
Codes over Finite Fields for Multidimensional Signals

AbstractIn this paper, linear block codes over finite fields of the algebraic integer ring of the cyclotomic field Q(e2πi/n) modulo irreducible elements are presented, where n∈{3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84}. These codes can be used for coding over ϕ(n) dimensional signal space and can correct one error taken from the group of all roots of unity in the algebraic integer ring of Q(e2πi/n). These codes provide an algebraic approach in an area which is currently mainly dominated by nonalgebraic convolutional codes
Integrable Roots in Split Graded Lie Algebras

AbstractA Lie algebra g is said to be split graded if it is graded by a torsion free abelian group Q in such a way that the subalgebra g0 is abelian and the operators adg0 are diagonalized by the grading. The elements of Q%{0} with gα≠{0} are called roots and a root α is said to be integrable if there are root vectors x±α∈g±α which are ad-nilpotent and generate an sl2-subalgebra g(α). In this paper we study subalgebras gΠ⊆g generated by the subalgebras g(α), α∈Π, where Π is a set of integrable roots. For |Π|≤2, these subalgebras are essentially Kac–Moody algebras which permits us to generalize several results on root strings from Kac–Moody algebras to split graded algebras. A central result is the local finiteness theorem saying that whenever all roots of a split graded Lie algebra μ are integrable, then μ is locally finite. If differences of roots in Π are not roots, then Π is called a simple system. In this case we describe the structure of the subalgebras gΠ in their relationship to the corresponding Kac–Moody algebras
Minimal identities for right-symmetric algebras

An algebra A with multiplication A × A → A, (a, b) → a ∘ b, is called right-symmetric, if a ∘ (b ∘ c) − (a ∘ b) ∘ c = a ∘ (c ∘ b) − (a ∘ c ) ∘ b, for any a, b, c ε A. The multiplication of right-symmetric Witt algebras Wn = {d∂i : u ε U, U = K[x1±1, …, x±1n], or = K[x1, …, xn], i = 1, …, n},p = 0, or W(m) = {u∂i :u ε U, U = On.(m)}, p > 0, are given by u∂i ∘ u∂j = ν∂j(u)∂i. An analogue of the Amitsur-Levitzki theorem for right-symmetric Witt algebras is established. Right-symmetric Witt algebras of rank n satisfy the standard right-symmetric identity of degree 2n + 1 : ΣσεSym2n sign(σ)aσ(1) ∘ (aσ(2) ∘ … ∘ (aσ(2n) ∘ a2n+1) …) = 0. The minimal degree for left polynomial identities of Wnr sym, Wn+r sym, p = 0, is 2n + 1. All left polynomial identities of right-symmetric Witt algebras of minimal degree follow from the left standard right-symmetric identity S2nr sym = 0, if p ≠ 2
Restrictions of representations of a surface group to a pair of free subgroups

Let Π be the fundamental group of a compact orientable genus m surface, and let G be a connected reductive algebraic group over an algebraically closed field of characteristic zero. Define two free rank m subgroups of Π by A = 〈a1, …, am〉 and B = 〈b1, …, bm〉, where Π = 〈a1, …, am, b1, …, bm |Πj=1m[aj, bj]〉 is the standard presentation of Π. We consider representations of Π, of A and of B into G. Restriction of representations induces a morphism from C(Π, G), the variety of closed conjugacy classes of representations of Π, to C(A, G)C(B, G). We prove that if m is greater than the semisimple rank of G then this morphism is dominant and almost all fibers are finite
Generalized Cocycles with Values in One-Units of Henselian Valued Division Algebras

AbstractLet Z/F be an inertial Galois extension of Henselian valued fields, and let D be a Z-central division algebra. Let G be a finite group acting on Z with fixed field F. We show that every generalized cocycle of G with values in the one-units of D is cohomologous to one of the form (θ,1), or in other words, the existence of such a cocycle implies that the group action of G on Z extends to a group action on D. We provide applications to lifting of group actions on the residue division algebra and to the existence of Kummer subfields in D given suitable data in [formula]
Invariants of universal enveloping algebras of relatively free lie algebras

Let FmW be the relatively free algebra of rank m ≥ 2 in the nonlocally nilpotent variety W of Lie algebras over an infinite field of any characteristic. We study the problem of finite generation of the algebra of invariants of a cyclic linear group G = 〈g〉 of finite order invertible in the base field, acting on the universal enveloping algebra U(FmW). If the matrix g has eigenvalues of different multiplicative orders, then we show that the algebra of invariants U(FmW)G is not finitely generated. If all eigenvalues of g ate of the same order and W is a subvariety of the variety RcU of all nilpotent of class c-by-abelian algebras for some c ≥ 1, then the algebra of invariants is finitely generated. On the other hand, for every g which is not a scalar matrix, there exists a variety of Lie algebras W such that the algebra U(FmW)G is not finitely generated
Closed asymptotic couples

The derivation of a Hardy field induces on its value group a certain function Ψ If a Hardy field extends the real field and is closed under powers, then its value group is also a vector space over ℝ. Such “ordered vector spaces with Ψ-function” are called H-couples. We define closed H-couples and show that every H-couple can be embedded into a closed one. The key fact is that closed H-couples have an elimination theory: solvability of an arbitrary system of equations and inequalities (built up from vector space operations, the function Ψ, parameters, and the unknowns to be solved for) is equivalent to an effective condition on the parameters of the system. The H-couple of a maximal Hardy field is closed, and this is also the case for the H-couple of the field of logarithmic-exponential series over ℝ. We analyze in detail finitely generated extensions of a given H-couple
Alexander Duality for Stanley–Reisner Rings and Squarefree Nn-Graded Modules

AbstractLet S=k[x1,…,xn] be a polynomial ring, and let ωS be its canonical module. First, we will define squarefreeness for Nn-graded S-modules. A Stanley–Reisner ring k[Δ]=S/IΔ, its syzygy module Syzi(k[Δ]), and ExtiS(k[Δ],ωS) are always squarefree. This notion will simplify some standard arguments in the Stanley–Reisner ring theory. Next, we will prove that the i-linear strand of the minimal free resolution of a Stanley–Reisner ideal IΔ⊂S has the “same information” as the module structure of ExtiS(k[Δ∨],ωS), where Δ∨ is the Alexander dual of Δ. In particular, if k[Δ] has a linear resolution, we can describe its minimal free resolution using the module structure of the canonical module of k[Δ∨], which is Cohen–Macaulay in this case. We can also give a new interpretation of a result of Herzog and co-workers, which states that k[Δ] is sequentially Cohen–Macaulay if and only if IΔ∨ is componentwise linear
Non-classical trigonal curves

We study trigonal Gorenstein curves of arithmetic genus g ≥ 5 such that g⅓ has a base point, through realizing them as canonical curves lying over a cone. Using an explicit description of such curves we compute the dimension of their moduli spaces. We also investigate non-classical trigonal Gorenstein curves; in fact, we give a complete classification of such curves for arithmetic genus g = 5. More generally, we classify the non-classical curves when the characteristic of the constant field is g - 1, g - 2, or 2g - 3. In characteristic 2 we also solve the case g = 2n + 1
A variation on a theme of Vasconcelos

A classical result of W. Vasconcelos (1967, J. Algebra 6, 309–316) and D. Ferrand (1967, C. R. Acad. Sci. Paris 264, 427–428) provides necessary and sufficient conditions for an ideal to be generated by a regular sequence. In this paper, we provide generally more accessible conditions guaranteeing this property, as well as some examples indicating that our hypotheses cannot be significantly weakened. The key step in our development is the use of lifting, as presented by M. Auslander, et al. (1993, J. Algebra 156, 273–317)
Hilbert functions for two ideals

Let (R, m) denote a Noetherian, local ring R with maximal ideal m. Let I and J be ideals contained in m and assume I + J is m-primary. Then for all non-negative integers r and s, the R-module [R/(I′ + Js)] has finite length. We denote the length of this R-module by lR/(I′ + Js)] = λ(r, s). The function λ(r, s) is called the Hilbert function of I and J. Let ℤ denote the integers and for p ε ℤ, let ℤ[≥ p] = {n ε ℤ|n ≥ p}. In this paper, we prove the following theorem: Suppose for some non-negative integer p, there exist ordered pairs (r1, s1), …, (rn, sn) ε ℤ[≥ p] × ℤ[≥ p] such that I′ ∩ Js ⊆ [Σi=1n(Ir−ri Js−si)(Iri ∩ Jsusi)] + (Ir+1 + Js+1) for all r, s ≥ p. Then there exists a polynomial f(x, y) ε of ℚ[x, y] (ℚ the rational numbers) such that λ(r, s) = f(r, s) for all r, s ≫ 0. Furthermore, ∂x(f) = dim(R/J), ∂y(f) = dim(R/I), and ∂(f) ≤ l(I) + l(J). Here ∂x(f)[∂y(f)] denotes the degree of f as a polynomial in x[y] and ∂(f) denotes the total degree of f. dim(S) is the Krull dimension of the ring S and l(U) is the analytical spread of the ideal U
Generic 2 × 2 matrices in positive characteristic

Let Rm(K) be the K-algebra generated by the generic 2 × 2 matrices y1, …, ym over a unitary commutative associative ring K. Our main result is that, with a small class of exceptions, for m a positive integer and p a prime, the kernel of the natural homomorphism Rm(ℤ) → Rm(ℤp) coincides with pRm(ℤ). The only exceptions are for m ≥ 5 and p = 2 when we give an explicit multilinear polynomial identity of degree 5 for the matrix algebra M2(ℤ2) which does not follow from the spolynomial identities of M2(ℤ2). This improves on Schelter's construction of a non-multilinear identity of this sort of degree 6, and Drensky and Tsiganchev's existence result for a multilinear identity such as we have found
Interpolation Domains

AbstractCall a domain D with quotient field K an interpolation domain if, for each choice of distinct arguments a1,…,an and arbitrary values c1,…,cn in D, there exists an integer-valued polynomial f (that is, f∈K[X] with f(D)⊆(D)), such that f(ai)=ci for 1≤i≤n. We characterize completely the interpolation domains if D is Noetherian or a Prüfer domain. In the first case, we show that D is an interpolation domain if and only if it is one-dimensional, locally unibranched with finite residue fields, in the second one, if and only if the ring Int(D)={f∈K[X]|f(D)⊆D} of integer-valued polynomials is itself a Prüfer domain. We also show that an interpolation domain must satisfy a double-boundedness condition, and thereby simplify a recent characterization of the domains D such that Int(D) is a Prüfer domain
Semisimple Orbits of Lie Algebras and Card-Shuffling Measures on Coxeter Groups

AbstractRandom walk on the chambers of hyperplane arrangements is used to define a family of card shuffling measures HW,x for a finite Coxeter group W and real x≠0. By algebraic group theory, there is a map Φ from the semisimple orbits of the adjoint action of a finite group of Lie type on its Lie algebra to the conjugacy classes of the Weyl group. Choosing such a semisimple orbit uniformly at random thereby induces a probability measure on the conjugacy classes of the Weyl group. For types A, B, and the identity conjugacy class of W for all types, it is proved that for q very good, this measure on conjugacy classes is equal to the measure arising from HW,q. The possibility of refining Φ to a map to elements of the Weyl group is discussed
Kawanaka Invariants for Representations of Weyl Groups

AbstractLet W be a Weyl group and let V be the natural CW-module, i.e., the reflection representation. For a complex irreducible character χ of W, we consider the invariantIχ;q≔|W|−1∑w∈Wχw2det1+qw|V/det1−qw|Vintroduced by N. Kawanaka. We determine I(χ;q) explicitly. Looking over these results, we observe a relation between Kawanaka's invariants I(χ;q) and the two-sided cells. For example, if a two-sided cell consists of a single element χ, then the Kawanaka invariant I(χ;q) can be expressed as ∏li=1(1+qhi)/(1−qhi) with some integers hi. This expression can be regarded as a quantization of the usual hook formula for the dimension of irreducible representations of symmetric groups
Splittable Lie Groups and Lie Algebras

AbstractWe show that each Mal'cev splittableK-Lie algebra g (i.e., each K-Lie algebra where ad(g) is splittable) with charK=0 may be realized as a splittable subalgebra of a gl(V), where V is a finite-dimensional vector space over K, and that each Mal'cev splittable analytic subgroup of a GL(n,R), i.e., each subgroup whose Lie algebra is Mal'cev splittable, can be realized as an analytic splittable subgroup of a GL(m,R). Finally these results are applied to establish a necessary and sufficient condition for the exponential function of a Mal'cev splittable Lie group to be surjective
The Principal 3-Blocks of Four- and Five-Dimensional Projective Special Linear Groups in Non-defining Characteristic

AbstractIn representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He has conjectured that, for any prime p, if a finite group G has an abelian Sylow p-subgroup P, then the principal p-blocks of G and the normalizer NG(P) of P in G are derived equivalent. Let q be a power of a prime such that q≡2 or 5 (mod9). In this paper we show that Broué's conjecture is true for p=3 and for G=PSL4(q) and G=PSL5(q). In these cases, G has elementary abelian Sylow 3-subgroups of order 9. What we prove here is the following. In the case G=PSL4(q) all the principal 3-blocks of G are Morita (even Puig) equivalent independently of infinitely many q. In the case G=PSL5(q) all the principal 3-blocks of G are Morita (even Puig) equivalent to the principal 3-block of NG(P) independently of infinitely many q
On the Bass–Lubotzky Question about Quotients of Hyperbolic Groups

AbstractWe prove at Theorem 1 that any non-elementary hyperbolic group G possesses a non-trivial finitely presented quotient Q having no non-trivial subgroups of finite indices. The theorem was “commissioned” in August 1997 by H. Bass and A. Lubotzky because the statement was required to constructing of their counter-examples to Platonov's conjecture on the representation rigid linear groups [BL]. Theorem 2 gives three equivalent reformulations of the known problem on existence of non-residually finite hyperbolic groups
Permutability Class of a Semigroup

AbstractGiven a permutation σ∈Sn, a semigroup S is called σ-permutable if x1x2···xn=xσ(1)xσ(2)···xσ(n) for all n-tuples x1, x2,…,xn∈S. Various permutability conditions were considered for special classes of permutations. We define the permutability class of a semigroup S to be the infinite sequence (G2, G3, G4,…) of the invariance groups Gn=Gn(S) of the operation x1x2···xn on S. The aim of this paper is to give a description of such sequences, which generalizes a number of old results on permutability conditions in semigroups
A Frobenius Formula for the Characters of Ariki–Koike Algebras

AbstractLet Hn,r be the Ariki–Koike algebra associated to the complex reflection group Wn,r=G(r,1,n). In this paper, we give a new presentation of Hn,r by making use of the Schur–Weyl reciprocity for Hn,r established by M. Sakamoto and T. Shoji (1999, J. Algebra, 221, 293–314). This allows us to construct various non-parabolic subalgebras of Hn,r. We construct all the irreducible representations of Hn,r as induced modules from such subalgebras. We show the existence of a partition of unity in Hn,r, which is specialized to a partition of unity in the group algebra CWn,r. Then we prove a Frobenius formula for the characters of Hn,r, which is an analogy of the Frobenius formula proved by A. Ram (1991, Invent. Math.106, 461–488) for the Iwahori–Hecke algebra of type A
Finite, Tame, and Wild Actions of Parabolic Subgroups in GL(V) on Certain Unipotent Subgroups

AbstractLet P be a parabolic subgroup of some general linear group GL(V) where V is a finite-dimensional vector space over an infinite field. The group P acts by conjugation on its unipotent radical Pu and via the adjoint action on pu, the Lie algebra of Pu. More generally, we consider the action of P on the lth member of the descending central series of pu, denoted by p(l)u. Let ℓ(pu) denote the nilpotency class of Pu. In our main result we show that P acts on p(l)u with a finite number of orbits precisely when ℓ(pu)≤4 for l=0, or ℓ(pu)≤5+2l for l≥1. Moreover, in case the field is algebraically closed, we consider the modality mod(P:p(l)u) of the action of P on p(l)u. We show that mod(P:p(l)u grows linearly in the minimal cases which admit infinitely many orbits (i.e., ℓ(pu)=5 for l=0, or ℓ(pu)=6+2l for l≥1), whereas the corresponding modality grows quadratically in all other infinite cases. These results are obtained by interpreting the orbits of P on p(l)u as isomorphism classes of good modules over certain quasi-hereditary algebras and by a detailed inspection of the Δ-representation types of these algebras
On the Irreducible Representations of the Specializations in Characteristics 2 and 3 of the Generic Hecke Algebra of Type F4

AbstractA complete determination of the irreducible modules of specialized Hecke algebras of type F4, with respect to specializations with equal parameters, has been obtained by M. Geck and K. Lux (1991, Manuscripta Math.70, 285–306) for all characteristics. A similar determination for specializations with v=u2 and v=u4 has been obtained by K. Bremke (1994, Manuscripta Math.83, 331–346). In another paper (1999, J. Algebra, 218, 654–671), the authors determined the irreducible modules for all remaining specializations other than those into fields of characteristic 2 or 3, obtaining en route decompositions of the generic irreducible modules under such specializations. In this paper, the corresponding results for characteristic 2 or 3 are obtained. Again, it is found that the decomposition matrices may be expressed in lower uni-triangular form in all these cases and that the splitting fields are those generated by the images of the parameters