Recently, there has been considerable progress in classifying the irreducible representations of Iwahori–Hecke algebras at roots of unity. Here, we present an application of these results to l-modular Harish–Chandra series for a finite group of Lie type G(q). Under some mild condition on l, we show that the l-modular principal series representations of G(q) are naturally parametrized by a subse...

the vertex arboricity $rho(g)$ of a graph $g$ is the minimum number of subsets into which the vertex set $v(g)$ can be partitioned so that each subset induces an acyclic graph‎. ‎a graph $g$ is called list vertex $k$-arborable if for any set $l(v)$ of cardinality at least $k$ at each vertex $v$ of $g$‎, ‎one can choose a color for each $v$ from its list $l(v)$ so that the subgraph induced by ev...

In this paper, I will provide a detailed explanation of Milnor’s construction of exotic 7-spheres. The candidate manifolds will be constructed as total spaces of S3 bundles over S4, denoted Mh,l. The subset of these candidates satisfying the condition h + l = ±1 will be shown to be topological spheres by Morse Theory. A subset of these that do not satisfy (h−l)2 ≡ 1 (mod 7) will be shown to not...

1. Dictatorship: The case where the value of f depends on a single Boolean variable (i.e., f(x) = xi ⊕ σ for some i ∈ [l] and σ ∈ {0, 1}). 2. Junta (of size k): The case where the value of f depends on at most k Boolean variables (i.e., f(x) = f (xI) for some k-subset I ⊂ [l] and f ′ : {0, 1}k → {0, 1}). 3. Monomial (of size k): The case where the value of f is the conjunction of exactly k Bool...

We continue the study of freezing sets for digital images introduced in [L. Boxer and P.C. Staecker, Fixed point topology, 1, Applied General Topology 2020; L. Boxer, 2, Convexity Freezing Sets Digital Topology, 2021]. prove methods obtaining $(X,c_i)$ $X \subset \mathbb{Z}^2$ $i \in \{1,2\}$. give examples to show how these can lead determination minimal sets.

In this paper we study some properties of the torsion function with Robin boundary conditions. Here write shape derivative $L^{\\infty}$ and $L^p$ norms, for $p\\ge 1$, function, seen as a functional on bounded simply connected open set $\\Omega\\subset\\mathbb{R}^n$, prove that balls are critical shapes these functionals, when volume $\\Omega$ is preserved.

Take G to be a split, simply-connected semisimple algebraic group (for example, SL(w, K) or Sp(2m, K)) over an algebraic closure K of the field of p elements, where p is prime. Denote by G(n) the finite subgroup consisting of elements of G defined over the field of p elements. The simple (rational) G-modules L(X) are parametrized by the set X of dominant weights, a subset of the character group...