Equivariant multiplicities of simply-laced type flag minors

Let g \mathfrak {g} be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D ¯ \overline {D} on the coordinate ring C [ N ] \mathbb {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 \mathfrak {g} is of type A n A_n or D 4 D_4 , the map D ¯ \overline {D} takes similar distinguished values on the set of all flag minors of C [ N ] \mathbb {C}[N] , raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D ¯ \overline {D} on flag minors belonging to the same standard seed, and we show that in any A D E 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 ] \mathbb {C}[N] via representations of quiver Hecke algebras.

Mirković–Vilonen basis in type 𝐴₁

Let G G be a connected reductive algebraic group over C \mathbb C . Through the geometric Satake equivalence, the fundamental classes of the Mirković–Vilonen cycles define a basis in each tensor product V ( λ 1 ) ⊗ ⋯ ⊗ V ( λ r ) V(\lambda _1)\otimes \cdots \otimes V(\lambda _r) of irreducible representations of G G . We compute this basis in the case G = S L 2 ( C ) G=\mathrm {SL}_2(\mathbb C) and conclude that in this case it coincides with the dual canonical basis at q = 1 q=1 .

Mixed QCD-EW corrections for Higgs leptonic decay via HW+W− vertex

We consider the two-loop corrections to the HW+W− vertex at order ααs. We construct a canonical basis for the two-loop integrals using the Baikov representation and the intersection theory. By solving the ϵ-form differential equations, we obtain fully analytic expressions for the master integrals in terms of multiple polylogarithms, which allow fast and accurate numeric evaluation for arbitrary configurations of external momenta. We apply our analytic results to the decay process H → νeeW, and study both the integrated and differential decay rates. Our results can also be applied to the Higgs production process via W boson fusion.

Robinson–Schensted–Knuth correspondence in the representation theory of the general linear group over a non-archimedean local field

We construct new “standard modules” for the representations of general linear groups over a local non-archimedean field. The construction uses a modified Robinson–Schensted–Knuth correspondence for Zelevinsky’s multisegments. Typically, the new class categorifies the basis of Doubilet, Rota, and Stein (DRS) for matrix polynomial rings, indexed by bitableaux. Hence, our main result provides a link between the dual canonical basis (coming from quantum groups) and the DRS basis.

Canonical bases arising from quantum symmetric pairs of Kac–Moody type

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$ -canonical bases for the highest weight integrable $\textbf U$ -modules and their tensor products regarded as $\textbf {U}^\imath$ -modules, as well as an $\imath$ -canonical basis for the modified form of the $\imath$ -quantum group $\textbf {U}^\imath$ . A key new ingredient is a family of explicit elements called $\imath$ -divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$ . We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi- $K$ -matrix and the constructions of $\imath$ -canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

Incremental Method of Generating Decision Implication Canonical Basis

Decision implication is an elementary representation of decision knowledge in formal concept analysis. Decision implication canonical basis (DICB), a set of decision implications with completeness and nonredundancy, is the most compact representation of decision implications. The method based on true premises (MBTP) for DICB generation is the most efficient one at present. In practical applications, however, data is always changing dynamically, and MBTP has to re-generate inefficiently the whole DICB. This paper proposes an incremental algorithm for DICB generation, which obtains a new DICB just by modifying and updating the existing one. Experimental results verify that when the samples in data are much more than condition attributes, which is actually a general case in practical applications, the incremental algorithm is significantly superior to MBTP. Furthermore, we conclude that, even for the data in which samples is less than condition attributes, when new samples are continually added into data, the incremental algorithm must be also more efficient than MBTP, because the incremental algorithm just needs to modify the existing DICB, which is only a part of work of MBTP.

The connectome spectrum as a canonical basis for a sparse representation of fast brain activity

ABSTRACTThe functional organization of neural processes is constrained by the brain’s intrinsic structural connectivity. Here, we explore the potential of exploiting this structure in order to improve the signal representation properties of brain activity and its dynamics. Using a multi-modal imaging dataset (electroencephalography, structural MRI and diffusion MRI), we represent electrical brain activity at the cortical surface as a time-varying composition of harmonic modes of structural connectivity. The harmonic modes are termed connectome harmonics, and their representation is known as the connectome spectrum of the signal. We found that: first, the brain activity signal is more compactly represented by the connectome spectrum than by the traditional area-based representation; second, the connectome spectrum characterizes fast brain dynamics in terms of signal broadcasting profile, revealing different temporal regimes of integration and segregation that are consistent across participants. And last, the connectome spectrum characterises fast brain dynamics with fewer degrees of freedom than area-based signal representations. Specifically, we show that with the connectome spectrum representation, fewer dimensions are needed to capture the differences between low-level and high-level visual processing, and the topological properties of the signal. In summary, this work provides statistical, functional and topological evidence supporting that by accounting for the brain’s structural connectivity fosters a more comprehensive understanding of large-scale dynamic neural functioning.