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Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander's correspondence, classical Schur–Weyl duality and Soergel's Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B, and module categories of endomorphism algebras of generator-relative cogenerators over B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n so that the endomorphism algebra of a generator-cogenerator provides an n-faithful cover, in the sense of Rouquier, of B. The degree n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple modules of B. We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category \mathcal {O} establishing an integral version of Soergel's Struktursatz. We show that deformations of the combinatorial Soergel's functor have better homological properties than the classical one. [ABSTRACT FROM AUTHOR]

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

Let X be a complex Banach space, and denote by \mathcal {B}(X) the algebra of all bounded linear operators on X. Let C,D\in \mathcal {B} \left (X\right) be fixed operators. In this paper, we characterize linear, continuous and bijective maps \varphi and \psi on \mathcal {B}\left (X\right) for which there exist invertible operators T_0, W_0 \in \mathcal { B}(X) such that \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that \varphi \left (A\right) \psi \left (B\right) =D in \mathcal {B}(X) whenever AB=C in \mathcal {B}(X). As a corollary, we deduce the form of linear, bijective and continuous maps \varphi on \mathcal {B}(X) having the property that \varphi \left (A\right) \varphi \left (B\right) =D in \mathcal {B}(X) whenever AB=C. [ABSTRACT FROM AUTHOR]

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer's cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a "cellular algebra Clifford theory" for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(\ell,p,n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(\ell,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck's result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(\ell,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of "adjustment matrices" in positive characteristic. [ABSTRACT FROM AUTHOR]

Let \Lambda (f) = K[x][y; f\frac {d}{dx} ] be an Ore extension of a polynomial algebra K[x] over a field K of characteristic zero where f\in K[x]. For a given polynomial f, the automorphism group of the algebra \Lambda (f) is explicitly described. The polynomial case \Lambda (0) = K[x,y] and the case of the Weyl algebra A_1= K[x][y; \frac {d}{dx} ] were done by Jung [J. Reine Angew. Math. 184 (1942), pp. 161–174] and van der Kulk [Nieuw Arch. Wisk. (3) 1 (1953), pp. 33–41], and Dixmier [Bul. Soc. Math. France 96 (1968), pp. 209–242], respectively. Alev and Dumas [Comm. Algebra 25 (1997), pp. 1655–1672] proved that the algebras \Lambda (f) and \Lambda (g) are isomorphic iff g(x) = \lambda f(\alpha x+\beta) for some \lambda, \alpha \in K\backslash \{ 0\} and \beta \in K. Benkart, Lopes and Ondrus [Trans. Amer. Math. Soc. 367 (2015), pp. 1993–2021] gave a complete description of the set of automorphism groups of algebras \Lambda (f). In this paper we complete the picture, i.e. given the polynomial f we have the explicit description of the automorphism group of \Lambda (f). The key concepts in finding the automorphism groups are the eigenform, the eigenroot and the eigengroup of a polynomial (introduced in the paper; they are of independent interest). [ABSTRACT FROM AUTHOR]

Let \Gamma \subseteq \mathbb {N} be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of \Gamma which depends only on the width of \Gamma, that is, the difference between the largest and the smallest generator of \Gamma. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width. [ABSTRACT FROM AUTHOR]

Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of product and permutation categories (PROPs) and product and braid categories (PROBs) with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action. [ABSTRACT FROM AUTHOR]

We describe some recent computer investigations with the 'Constraint Logic Programming over Finite Domains' library in the Prolog programming environment to search for new simple Lie algebras over the field \operatorname {GF}(2) of 2 elements. Motivated by a paper of Grishkov et al., we specifically look for those with a thin decomposition , and we settle one of their conjectures. We extrapolate from our results the existence of two new infinite families of simple Lie algebras, in addition to finding seven new sporadic examples in dimension 31. We also better contextualise some previously discovered simple algebras, putting them into families which do not seem to have ever appeared in the literature, and give an updated table of those currently known. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a poset isomorphism between two support \tau-tilting posets. We take several information from combinatorial properties of support \tau-tilting posets and give an analogue of Happel-Unger's reconstruction theorem. [ABSTRACT FROM AUTHOR]

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C^\ast-algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C^\ast-algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space. [ABSTRACT FROM AUTHOR]

In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form. [ABSTRACT FROM AUTHOR]

Let (A,\mathfrak {m}) be an excellent normal domain of dimension two containing a field k \cong A/\mathfrak {m}. An \mathfrak {m}-primary ideal I is a p_g-ideal if the Rees algebra A[It] is a Cohen-Macaulay normal domain. If k is algebraically closed then Okuma, Watanabe and Yoshida proved that A has p_g-ideals and furthermore product of two p_g-ideals is a p_g ideal. Previously we showed that if k has characteristic zero then A has p_g-ideals. In this paper we prove that if k is perfect field of positive characteristic then also A has p_g ideals. [ABSTRACT FROM AUTHOR]

In this paper, for an arbitrary Kac-Moody Lie algebra {\mathfrak g} and a diagram automorphism \mu of {\mathfrak g} satisfying certain natural linking conditions, we introduce and study a \mu-twisted quantum affinization algebra {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) of {\mathfrak g}. When {\mathfrak g} is of finite type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is Drinfeld's current algebra realization of the twisted quantum affine algebra. When \mu =\mathrm {id} and {\mathfrak g} in affine type, {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) is the quantum toroidal algebra introduced by Ginzburg, Kapranov and Vasserot. As the main results of this paper, we first prove a triangular decomposition for {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Second, we give a simple characterization of the affine quantum Serre relations on restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules in terms of "normal order products". Third, we prove that the category of restricted {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right)-modules is a monoidal category and hence obtain a topological Hopf algebra structure on the "restricted completion" of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right). Last, we study the classical limit of {\mathcal U}_\hbar \left (\hat {\mathfrak g}_\mu \right) and abridge it to the quantization theory of extended affine Lie algebras. In particular, based on a classification result of Allison-Berman-Pianzola, we obtain the \hbar-deformation of all nullity 2 extended affine Lie algebras. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

Upper triangular, and more generally, block-triangular matrices, are rather important in Linear Algebra, and also in Ring theory, namely in the theory of PI algebras (algebras that satisfy polynomial identities). The group gradings on such algebras have been extensively studied during the last decades. In this paper we prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. As noted by F. Yasumura [Arch. Math. (Basel) 110 (2018), pp. 327–332] this yields the classification of the group gradings on these algebras and confirms a conjecture made by A. Valenti and M. Zaicev [Arch. Math. (Basel) 89 (2007), pp. 33–40]. [ABSTRACT FROM AUTHOR]

We equip the type A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p it becomes a p-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. We conjecture that the p-dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p-canonical basis. More precise conjectures will be found in the sequel. Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A. We also examine a particular Bott-Samelson bimodule in type A_7, which is indecomposable in characteristic 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p-dg setting of being indecomposable. [ABSTRACT FROM AUTHOR]

Let \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of \omega-semistable properly supported sheaves on Y with fixed slope \mu and \zeta-semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on \zeta depending on the polarization \omega and \mu. These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of \omega-semistable properly supported sheaves on Y with fixed slope \mu: for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams. [ABSTRACT FROM AUTHOR]

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic 0, the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras. [ABSTRACT FROM AUTHOR]

Let \mathcal {B}(H) be the algebra of all bounded operators on a Hilbert space H. Let T=V|T| be the polar decomposition of an operator T\in \mathcal {B}(H). The mean transform of T is defined by M(T)=\frac {T+|T|V}{2}. In this paper, we discuss several properties related to the spectrum, the kernel, the image, and the polar decomposition of mean transform. Moreover, we investigate the image and preimage by the mean transform of some class of operators such as positive, normal, unitary, hyponormal, and co-hyponormal operators. [ABSTRACT FROM AUTHOR]

The heat semigroup on discrete hypercubes is well-known to be contractive over L_p-spaces for 1

In this paper, we give an explicit realization of all irreducible modules in Chari's category \widetilde {\mathcal O} for the affine Kac-Moody algebra A_{1}^{(1)} by using the idea of free fields. We work on a much more general setting which also gives us explicit realizations of all simple weight modules for certain current algebra of \mathfrak {sl}_2(\mathbb {C}) with finite weight multiplicities, including the polynomial current algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t], the loop algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1}] and the three-point Lie algebra \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1},(t-1)^{-1}] arisen in the work by Kazhdan-Lusztig. [ABSTRACT FROM AUTHOR]

Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformation \tau _b of the generating series of bipartite maps, which generalizes the partition function of \beta-ensembles of random matrices. The Matching-Jack conjecture suggests that the coefficients c^\lambda _{\mu,\nu } of the function \tau _b in the power-sum basis are non-negative integer polynomials in the deformation parameter b. Dołęga and Féray have proved in 2016 the "polynomiality" part in the Matching-Jack conjecture, namely that coefficients c^\lambda _{\mu,\nu } are in \mathbb {Q}[b]. In this paper, we prove the "integrality" part, i.e. that the coefficients c^\lambda _{\mu,\nu } are in \mathbb {Z}[b]. The proof is based on a recent work of the author that deduces the Matching-Jack conjecture for marginal sums \overline { c}^\lambda _{\mu,l} from an analog result for the b-conjecture, established in 2020 by Chapuy and Dołęga. A key step in the proof involves a new connection with the graded Farahat-Higman algebra. [ABSTRACT FROM AUTHOR]

In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to the Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in \mathbb {R}^4, which can be finitely presented algebraically as a circuit algebra , or equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups \mathsf {KV} and \mathsf {KRV}—the symmetry groups of Kashiwara-Vergne solutions—as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebras of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmüller group \mathsf {GRT}_1 as automorphisms of arrow diagrams. [ABSTRACT FROM AUTHOR]

Let B be an integral domain over a field K of characteristic 0. The derivation δ of B[Yd] = B[y1, . . . ,yd] is elementary if δ (B) = 0 and δ (yi) ∈ B, i = 1, . . . ,d. Then for d ≥ 2 the elements uij = δ (yi)yj-δ (yj)yi, 1 ≤ i < j ≤ d, belong to the algebra B[Yd]δ of constants of δ, and it is a natural question whether the B-algebra B[Yd]δ is generated by all uij. In this paper we consider the special case of B = K[Xd] = K[x1, . . . ,xd]. If δ (yi) = xi, i = 1, . . . ,d, this is the Nowicki conjecture from 1994, which was confirmed in several papers based on different methods. The case δ (yi) = xini, ni > 0, i = 1, . . . ,d, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009, if δ (yi) = ƒi(xi), for any nonconstant polynomials ƒi(xi), i = 1, . . . ,d, then B[Yd]δ = K[Xd,Yd]δ is generated by Xd and Ud = uij = ƒi(xi)yj-yiƒj(xj) mid 1 ≤ i < j ≤ d. In the present paper we have found a presentation of the algebra K[Xd,Yd]δ = K[Xd,Ud mid R = S = 0],d ≥ 4, R = r(i,j,k,l)mid 1 ≥ i < j < k < l \≤ d, S = s(i,j,k)mid 1 ≤ i < j < k ≤ d, and a basis of K[Xd,Yd]δ as a vector space. As a corollary we have shown that the defining relations R ∩ S form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009. The algebras K[Xd,Yd]δ, d < 4, are also described. [ABSTRACT FROM AUTHOR]

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras. [ABSTRACT FROM AUTHOR]

Ritt's theorem of zeroes and Siedenberg's embedding theorem are classical results in differential algebra allowing to connect algebraic and model-theoretic results on nonlinear PDEs to the realm of analysis. However, the existing proofs of these results use sophisticated tools from constructive algebra (characteristic set theory) and analysis (Riquier's existence theorem). In this paper, we give new short proofs for both theorems relying only on basic facts from differential algebra and the classical Cauchy-Kovalevskaya theorem for PDEs. [ABSTRACT FROM AUTHOR]

In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded. [ABSTRACT FROM AUTHOR]

Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous-trace C^*-algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arises naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper. [ABSTRACT FROM AUTHOR]

Let \mathcal {A} denote a central hyperplane arrangement of rank n in affine space \mathbb {K}^n over a field \mathbb {K} of characteristic zero and let l_1,\ldots, l_m\in R≔\mathbb {K}[x_1,\ldots,x_n] denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial f≔l_1\cdots l_m\in R. The focus of the paper is on the ideal J_f\subset R generated by the partial derivatives of f. We conjecture that J_f is a minimal reduction of the ideal \mathbb {I}\subset R generated by the (m-1)-fold products of distinct forms among l_1,\ldots, l_m. We prove this conjecture for an almost generic \mathcal {A} (i.e., any n-1 among the defining linear forms are linearly independent). In this case we obtain a stronger version of a result by Dimca and Papadima, and we confirm the conjecture unconditionally for n=3. We also conjecture that J_f is an ideal of linear type (i.e., the respective symmetric and Rees algebras coincide). We prove this conjecture for n=3. In the sequel we explain the tight relationship between the two ideals J_f, \mathbb {I}\subset R; in particular, we show that in the generic case (J_f)^{\text {sat}}=\mathbb I. As a consequence, we can provide a simpler proof of a conjectured result of Yuzvinsky, proved by Rose and Terao, on the vanishing of the depth of R/J_f. [ABSTRACT FROM AUTHOR]

In this short note we correct the statement of the main result of [Trans. Amer. Math. Soc. 373 (2020), no. 3, 1909-1940]. That paper presented the rational cohomology ring of a toric arrangement by generators and relations. One of the series of relations given in the paper is indexed over the set circuits in the arrangement's arithmetic matroid. That series of relations should however be indexed over all sets X with |X| = rk(X)+1. Below we give the complete and correct presentation of the rational cohomology ring. [ABSTRACT FROM AUTHOR]

Majorana representations have been introduced by Ivanov [ Cambridge Tracts in Mathematics , Cambridge University Press, Cambridge, 2009] in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by Fischer involutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of A_{12} (by Franchi, Ivanov, and Mainardis [J. Algebraic Combin. 44 (2016), pp. 265-292], the largest alternating group admitting a Majorana representation) for this might eventually lead to a new and independent construction of the Monster group (see A.A Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, Section 4, page 115]). In this paper we prove that A_{12} has two possible Majorana sets, one of which is the set \mathcal X_b of involutions of cycle type 2^2, the other is the union of \mathcal X_b with the set \mathcal X_s of involutions of cycle type 2^6. The latter case (the saturated case) is most interesting, since the Majorana set is precisely the set of involutions of A_{12} that fall into the class of Fischer involutions when A_{12} is embedded in the Monster. We prove that A_{12} has a unique saturated Majorana representation and we determine its degree and decomposition into irreducibles. As consequences we get that the Harada-Norton group has, up to equivalence, a unique Majorana representation and every Majorana algebra, affording either a Majorana representation of the Harada-Norton group or a saturated Majorana representation of A_{12}, satisfies the Straight Flush Conjecture (see A. A. Ivanov [ Contemp. Math. , Amer. Math. Soc., Providence, RI, 2017, pp. 11-17] and A. A. Ivanov [ Group theory and computation , Indian Stat. Inst. Ser., Springer, Singapore, 2018, pp. 107-118]). As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on A_8, the four point stabilizer subgroup of A_{12}. We finally state a conjecture about Majorana representations of the alternating groups A_n, 8\leq n\leq 12. [ABSTRACT FROM AUTHOR]

In this paper, we show that the finite subalgebra \mathcal {A}^\mathbb {R}(1), generated by \mathrm {Sq}^1 and \mathrm {Sq}^2, of the \mathbb {R}-motivic Steenrod algebra \mathcal {A}^\mathbb {R} can be given 128 different \mathcal {A}^\mathbb {R}-module structures. We also show that all of these \mathcal {A}-modules can be realized as the cohomology of a 2-local finite \mathbb {R}-motivic spectrum. The realization results are obtained using an \mathbb {R}-motivic analogue of the Toda realization theorem. We notice that each realization of \mathcal {A}^\mathbb {R}(1) can be expressed as a cofiber of an \mathbb {R}-motivic v_1-self-map. The {\mathrm {C}_2}-equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the \mathrm {RO}({\mathrm {C}_2})-graded Steenrod operations on a {\mathrm {C}_2}-equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the {\mathrm {C}_2}-equivariant realizations of \mathcal {A}^{\mathrm {C}_2}(1). We find another application of the \mathbb {R}-motivic Toda realization theorem: we produce an \mathbb {R}-motivic, and consequently a {\mathrm {C}_2}-equivariant, analogue of the Bhattacharya-Egger spectrum \mathcal {Z}, which could be of independent interest. [ABSTRACT FROM AUTHOR]

For a primitive Dirichlet character \chi modulo q, we define M(\chi)=\max _{t } |\sum _{n \leq t} \chi (n)|. In this paper, we study this quantity for characters of a fixed odd order g\geq 3. Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character \chi we have \begin{equation*} M(\chi)\ll _{\varepsilon } \sqrt {q}(\log q)^{1-\delta _g}(\log \log q)^{-1/4+\varepsilon }, \end{equation*} where \delta _g ≔1-\frac {g}{\pi }\sin (\pi /g). This improves upon the works of Granville and Soundararajan [J. Amer. Math. Soc. 20 (2007), pp. 357–384] and of Goldmakher [Algebra Number Theory 6 (2012), pp. 123–163]. Furthermore, assuming the Generalized Riemann Hypothesis (GRH) we prove that \begin{equation*} M(\chi) \ll \sqrt {q} \left (\log _2 q\right)^{1-\delta _g} \left (\log _3 q\right)^{-\frac {1}{4}}\left (\log _4 q\right)^{O(1)}, \end{equation*} where \log _j is the j-th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of \log _4 q). One of the key ingredients in the proof of the upper bounds is a new Halász-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest. [ABSTRACT FROM AUTHOR]

In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra. [ABSTRACT FROM AUTHOR]

In this paper we give a version of Bergman's diamond lemma which applies to certain monoidal categories presented by generators and relations. In particular, it applies to: the Coxeter presentation of the symmetric groups, the quiver Hecke algebras of Khovanov-Lauda-Rouquier, the Webster tensor product algebras, and various generalizations of these. We also give an extension of Manin-Schechtmann theory to non-reduced expressions. [ABSTRACT FROM AUTHOR]

Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation \mathcal {A}: \mathbb {R}[t] \to \mathbb {R}[t] defined by \mathcal {A}(t^n) = A_n(t), where A_n(t) denotes the n-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator \mathcal {A}, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials. [ABSTRACT FROM AUTHOR]

In this paper we show that evolution algebras over any given field \Bbbk are universally finite. In other words, given any finite group G, there exist infinitely many regular evolution algebras X such that Aut(X)\cong G. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme \mathbf {G} over \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra. [ABSTRACT FROM AUTHOR]

Let g be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D on the coordinate ring C[N] related to Brion's equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g is of type An or D4, the map D takes similar distinguished values on the set of all flag minors of C[N], raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D on flag minors belonging to the same standard seed, and we show that in any ADE type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh's monoidal categorification of the cluster structure of C[N] via representations of quiver Hecke algebras. [ABSTRACT FROM AUTHOR]

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples. [ABSTRACT FROM AUTHOR]

In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if A is a quasi-hereditary algebra with a simple preserving duality and T is a faithful tilting A-module, then A has the double centralizer property with respect to T. This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module T over A for which A = EndEndA(T)(T). As an application, we establish a Schur-Weyl duality between the symplectic Schur algebra SKsy(m,n) and the Brauer algebra Bn(−2m) on the space of dual partially harmonic tensors under certain condition. [ABSTRACT FROM AUTHOR]

Zeta values in Tate algebras were introduced by Pellarin in 2012. They are generalizations of Carlitz zeta values and play an increasingly important role in function field arithmetic. In this paper we prove a conjecture of Pellarin on identities for these zeta values. The proof is based on arithmetic properties of Carlitz zeta values and an explicit formula for Bernoulli-type polynomials attached to zeta values in Tate algebras. [ABSTRACT FROM AUTHOR]

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians J for which the Galois group associated to their 2-torsion is insoluble and "large" (relative to the dimension of J). In this paper we examine what happens when this Galois group merely contains an element of "large" prime order. [ABSTRACT FROM AUTHOR]

In this paper, we follow an idea of Lusztig to define the Fourier transform on the Ringel-Hall algebra of a valued quiver (given by a quiver with automorphism). As an application, this provides a direct proof of the fact that the Ringel-Hall algebra of a valued quiver is independent of its orientation. Furthermore, by combining the BGP-reflection operators defined on double Ringel-Hall algebras of valued quivers with Fourier transforms, we obtain an alternative construction of Lusztig's symmetries of the associated quantum enveloping algebras. This generalizes a result of Sevenhant and Van den Bergh in the quiver case. [ABSTRACT FROM AUTHOR]

Consider a proper, isometric action by a unimodular, locally compact group G on a complete Riemannian manifold M. For equivariant elliptic operators that are invertible outside a cocompact subset of M, we show that a localised index in the K-theory of the maximal group C*-algebra of G is well-defined. The approach is based on the use of maximal versions of equivariant localised Roe algebras, and many of the technical arguments in this paper are used to handle the ways in which they differ from their reduced versions. By using the maximal group C*-algebra instead of its reduced counterpart, we can apply the trace given by integration over G to recover an index defined earlier by the last two authors, and developed further by Braverman, in terms of sections invariant under the group action. This leads to refinements of index-theoretic obstructions to Riemannian metrics of positive scalar curvature on noncompact manifolds, and also on orbifolds and other singular quotients of proper group actions. As a motivating application in another direction, we prove a version of Guillemin and Sternberg's quantisation commutes with reduction principle for equivariant indices of Spinc Callias-type operators. [ABSTRACT FROM AUTHOR]

It is natural to ask whether it is possible to construct Lévy-like processes where actions by random elements of a given semigroup play the role of increments. Such semigroups induce a convolution-like algebra structure in the space of finite measures. In this paper, we show that the Whittaker convolution operator, related with the Shiryaev process, gives rise to a convolution measure algebra having the property that the convolution of probability measures is a probability measure. We then introduce the class of Lévy processes with respect to the Whittaker convolution and study their basic properties. We obtain a martingale characterization of the Shiryaev process analogous to Lévy's characterization of Brownian motion. Our results demonstrate that a nice theory of Lévy processes with respect to generalized convolutions can be developed for differential operators whose associated convolution does not satisfy the usual compactness assumption on the support of the convolution. [ABSTRACT FROM AUTHOR]

Let R be a standard graded Gorenstein algebra over a field presented by quadrics. In [Compositio Math. 129 (2001), no. 1, 95-121], Conca-Rossi-Valla show that such a ring is Koszul if reg R ≤ 2 or if reg R = 3 and c = codim R ≤ 4, and they ask whether this is true for reg R = 3 in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring R that guarantee the Nagata idealization ~ R = R × ωR(−a−1) is a non-Koszul quadratic Gorenstein ring. We prove there exist rings of regularity 3 satisfying our conditions for all c ≥ 9; this yields a negative answer to the question from the above-mentioned paper. [ABSTRACT FROM AUTHOR]

In this paper, it is established that quantum nilpotent algebras (also known as CGL extensions) are catenary, i.e., all saturated chains of inclusions of prime ideals between any two given prime ideals P ⊊ Q have the same length. This is achieved by proving that the prime spectra of these algebras have normal separation, and then establishing the mild homological conditions necessary to apply a result of Lenagan and the first author. The work also recovers the Tauvel height formula for quantum nilpotent algebras, a result that was first obtained by Lenagan and the authors through a different approach. [ABSTRACT FROM AUTHOR]

Let g be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let μ be a diagram automorphism of g, and let L(g,μ) be the loop algebra of g associated to μ. In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension g[μ] of L(g,μ). The construction contains the classical limit of Drinfeld's new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212-216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283-307] as special examples. As an application, when g is of simply-laced-type, we prove that the classical limit of the μ-twisted quantum affinization of the quantum Kac-Moody algebra associated to g introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of g[μ]. [ABSTRACT FROM AUTHOR]

The simplest version of Bertini's irreducibility theorem states that the generic fiber of a noncomposite polynomial function is an irreducible hypersurface. The main result of this paper is its analog for a free algebra: if ƒ is a noncommutative polynomial such that ƒ-λ factors for infinitely many scalars λ, then there exist a noncommutative polynomial h and a nonconstant univariate polynomial p such that ƒ = p ○ h. Two applications of free Bertini's theorem for matrix evaluations of noncommutative polynomials are given. An eigenlevel set of ƒ is the set of all matrix tuples X where ƒ(X) attains some given eigenvalue. It is shown that eigenlevel sets of ƒ and g coincide if and only if ƒa = ag for some nonzero noncommutative polynomial a. The second application pertains to quasiconvexity and describes polynomials ƒ such that the connected component of X tuple of symmetric n × n matrices: λI ≻ ƒ(X) about the origin is convex for all natural n and λ > 0. It is shown that such a polynomial is either everywhere negative semidefinite or the composition of a univariate and a convex quadratic polynomial. [ABSTRACT FROM AUTHOR]