Recent zbMATH articles in MSC 20C08https://www.zbmath.org/atom/cc/20C082021-05-28T16:06:00+00:00WerkzeugTwo boundary centralizer algebras for \(\mathfrak{q}(n)\).https://www.zbmath.org/1459.170332021-05-28T16:06:00+00:00"Zhu, Jieru"https://www.zbmath.org/authors/?q=ai:zhu.jieruThe celebrated Schur-Weyl duality, which shows a double centralizer property between the symmetric group \(\mathfrak{S}_d\) and the general linear Lie algebra \(\mathfrak{gl}(V)\) on the tensor space \(V^{\otimes d}\), provides a powerful link between the representation theories of \(\mathfrak{S}_d\) and \(\mathfrak{gl}(V)\). Nowadays it has been generalized to many other settings.
Replacing \(V^{\otimes d}\) by \(M\otimes V^{\otimes d}\) where \(M\) is a finite dimensional \(\mathfrak{gl}(V)\)-module, Arakawa-Suzuki studied the action of the affine Hecke algebra \(\mathcal{H}_d^\mathrm{aff}\) which centralizes the action of \(\mathfrak{gl}(V)\). Furthermore, Daugherty studied the degenerate two-boundary braid algebra \(\mathcal{G}_d\) and its quotient \(\mathcal{H}_d^\mathrm{ext}\), both of which admit a well-defined action on \(M\otimes N\otimes V^{\otimes d}\) for irreducible \(\mathfrak{gl}(V)\)-modules \(M,N\) parameterized by rectangular Young diagrams.
A super analogue of Daugherty's two-boundary setting was developed by Zhu (the author of the paper under review) in an unpublished paper [Two boundary centralizer algebras for \(\mathfrak{gl}(n\,\vert\, m)\), \url{arXiv:1809.08172}
] where \(\mathcal{H}_d^\mathrm{ext}\) and \(\mathfrak{gl}_{n|m}\) form the duality.
On the other hand, Hill-Kujawa-Sussan generalized Arakawa-Suzuki's construction to the type Q version, where \(\mathcal{H}_d^\mathrm{aff}\) and \(\mathfrak{gl}(V)\) are replaced by the affine Hecke-Clifford algebra \(H_d\) and the queer superalgebra \(\mathfrak{q}(n)\), respectively.
In the paper under review, the author provides a generalization of Hill-Kujawa-Sussan's work to the two boundary setting. She defined the degenerate two boundary affine Hecke-Clifford algebra \(\mathcal{H}_d\) and its quotient \(\mathcal{H}_{p,d}\). This quotient \(\mathcal{H}_{p,d}\) admits a \(\mathfrak{q}(n)\)-linear action on \(M\otimes N\otimes V^{\otimes d}\) where \(V\) is the natural \(\mathfrak{q}(n)\)-module and \(M\) (resp. \(N\)) is a irreducible highest weight module parameterized by a staircase partition (resp. a single row). She shows that as \(\mathcal{H}_{p,d}\)-modules, the irreducible summands of \(M\otimes N\otimes V^{\otimes d}\) can be constructed explicitly by some combinational tools such as shifted tableaux and the Bratteli graph. These modules belong to a family of modules called calibrated.
Reviewer: Li Luo (Shanghai)Casselman's basis of Iwahori vectors and Kazhdan-Lusztig polynomials.https://www.zbmath.org/1459.220042021-05-28T16:06:00+00:00"Bump, Daniel"https://www.zbmath.org/authors/?q=ai:bump.daniel"Nakasuji, Maki"https://www.zbmath.org/authors/?q=ai:nakasuji.makiLet \(F\) be a nonarchimedean local \(p\)-adic field and \(G\) a split semisimple \(p\)-adic group over \(F\). Let \((\pi,V)\) be a spherical representation of G. Casselman defined a basis \(\{f_{w}\}_{w\in W}\) of Iwahori fixed vectors of spherical representation \((\pi,V)\) as being dual to the intertwining operators, where \(W\) is the Weyl group [\textit{W. Casselman}, Compos. Math. 40, 387--406 (1980; Zbl 0472.22004)]. On the other hand there is a natural basis \(\{\psi_{w}\}_{w\in W}\) and one seeks to find the transition matrices between the two bases. Casselman himself says that; it is an unsolved problem and, as I can see, a difficult one to express these bases in terms of one another. However, For his applications, which include the computation of the spherical function and some other cases it is only necessary to compute one element of the basis explicitly [\textit{W. Casselman} and \textit{J. Shalika}, Compos. Math. 41, 207--231 (1980; Zbl 0472.22005)].
The authors of the paper under review by using their other work [\textit{D. Bump} and \textit{M. Nakasuji}, Can. J. Math. 63, No. 6, 1238--1253 (2011; Zbl 1230.22009)] and express the transition matrix \((m_{u,v})\) of the Casselman basis to natural basis in terms of certain polynomials that are deformations of the Kazhdan-Lusztig R-polynomials. The paper is well written and finishes with a conjecture.
Reviewer: Manouchehr Misaghian (Prairie View)Quantum Schur duality of affine type C with three parameters.https://www.zbmath.org/1459.170302021-05-28T16:06:00+00:00"Fan, Zhaobing"https://www.zbmath.org/authors/?q=ai:fan.zhaobing"Lai, Chun-Ju"https://www.zbmath.org/authors/?q=ai:lai.chun-ju"Li, Yiqiang"https://www.zbmath.org/authors/?q=ai:li.yiqiang"Luo, Li"https://www.zbmath.org/authors/?q=ai:luo.li"Wang, Weiqiang"https://www.zbmath.org/authors/?q=ai:wang.weiqiang"Watanabe, Hideya"https://www.zbmath.org/authors/?q=ai:watanabe.hideyaSummary: We establish a three-parameter Schur duality between the affine Hecke algebra of type C and a coideal subalgebra of quantum affine \(\mathfrak{sl}_n\). At the equal parameter specializations, we obtain Schur dualities of types BCD.Tied links and invariants for singular links.https://www.zbmath.org/1459.570022021-05-28T16:06:00+00:00"Aicardi, F."https://www.zbmath.org/authors/?q=ai:aicardi.francesca"Juyumaya, J."https://www.zbmath.org/authors/?q=ai:juyumaya.jesusSummary: Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the tied braid monoid has a decomposition like a semi-direct group product. By using this decomposition we reprove the Alexander and Markov theorem for tied links; also, we introduce the tied singular knots, the tied singular braid monoid and certain families of Homflypt type invariants for tied singular links; these invariants are five-variables polynomials. Finally, we study the behavior of these invariants; in particular, we show that our invariants distinguish non isotopic singular links indistinguishable by the Paris-Rabenda invariant [\textit{L. Paris} and \textit{L. Rabenda}, Ann. Inst. Fourier 58, No. 7, 2413--2443 (2008; Zbl 1171.57008)].Mirabolic subgroup, refined Newton stratification and cohomology of Lubin-Tate spaces.https://www.zbmath.org/1459.111252021-05-28T16:06:00+00:00"Boyer, Pascal"https://www.zbmath.org/authors/?q=ai:boyer.pascalSummary: In [the author, Invent. Math. 177, No. 2, 239--280 (2009; Zbl 1172.14016)], we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle \(\Psi\) of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of \(\Psi \). In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.Morita equivalence for \(k\)-algebras.https://www.zbmath.org/1459.160032021-05-28T16:06:00+00:00"Aubert, Anne-Marie"https://www.zbmath.org/authors/?q=ai:aubert.anne-marie"Baum, Paul"https://www.zbmath.org/authors/?q=ai:baum.paul-f"Plymen, Roger"https://www.zbmath.org/authors/?q=ai:plymen.roger-j"Solleveld, Maarten"https://www.zbmath.org/authors/?q=ai:solleveld.maartenThe authors analyze the stratified equivalence between \(k\)-algebras, introduced in Section 9. This equivalence preserves the spectrum, the periodic cyclic holomogy, and results to be weaker than Morita equivalence: Morita equivalence implies stratified (Proposition 9.1) but stratified equivalence does not imply Morita (Theorem 9.3).
One advantage of this equivalence is that it works well for finite \(k\)-algebras, without regarding if the algebra is unital, unlike in the Morita context. In particular, a finite \(k\)-algebra is stratified equivalent to its algebra of matrices (Proposition 9.5).
In Section 10, they present some properties of this equivalence, for instance, it respects tensor product and crossed product. As an example, in Section 11, they find examples of stratified equivalent algebra that are not Morita equivalent over Hecke algebras.
For the entire collection see [Zbl 07265617].
Reviewer: Luz Adriana Mejia Castaño (Barranquilla)Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology.https://www.zbmath.org/1459.570182021-05-28T16:06:00+00:00"Gorsky, Eugene"https://www.zbmath.org/authors/?q=ai:gorsky.eugene"Neguţ, Andrei"https://www.zbmath.org/authors/?q=ai:negut.andrei"Rasmussen, Jacob"https://www.zbmath.org/authors/?q=ai:rasmussen.jacob-aThe paper under review constructs a categorification of the maximal commutative subalgebra of a type \(A\) Hecke algebra, spanned by the Jones-Wenzl projectors to irreducible subrepresentations of the regular representation. This categorification is geometric in nature and is constructed via a monoidal functor from the symmetric monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The latter monoidal category is a standard categorification of the Hecke algebra.
The adjoint of this functor allows one to relate the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme.
The general picture the authors produce leads to a number of conjectures. For example, one of these conjectures is that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. Several other conjecture can be found in the introduction.
Reviewer: Volodymyr Mazorchuk (Uppsala)