Let $S=K[x_1,\dots ,x_n]$ be the polynomial ring in n variables over a field K. In this paper, we compute the socle of c-bounded strongly stable ideals and determine the saturation numbers of strongly stable ideals and equigenerated c-bounded strongly stable ideals. We also provide explicit formulas for the saturation number $\mathrm{sat}(I)$ of Veronese-type ideals I. Using this formula, we show that $\mathrm{sat}(I^k)$ is quasilinear from the beginning, and we determine the quasilinear function explicitly.

Let $\mathrm{h}(\mathcal{C})$ be the homotopy category of a stable infinity category $\mathcal{C}$. Then the homotopy category $\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ of morphisms in the stable infinity category $\mathcal{C}$ is also triangulated. Hence, the space $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ of stability conditions on $\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ is well-defined though the nonemptiness of $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ is not obvious. Our basic motivation is a comparison of the homotopy type of $\mathsf{Stab}\mathrm{h}(\mathcal{C})$ and that of $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$. Under the motivation, we show that functors $d_0$ and $d_1:{\mathcal{C}}^{{\mathrm{\Delta }}^1}\rightrightarrows \mathcal{C}$ induce continuous maps from $\mathsf{Stab}\mathrm{h}(\mathcal{C})$ to $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$ contravariantly where $d_0$ (resp., $d_1$) takes a morphism to the target (resp., source) of the morphism. As a consequence, if $\mathsf{Stab}\mathrm{h}(\mathcal{C})$ is nonempty, then so is $\mathsf{Stab}\mathrm{h}({\mathcal{C}}^{{\mathrm{\Delta }}^1})$. Assuming $\mathcal{C}$ is the derived infinity category of the projective line over a field, we further study basic properties of $d_0^{\ast }$ and $d_1^{\ast }$. In addition, we give an example of a derived category which does not have any stability condition.

We prove a number of results surrounding the Borsuk–Ulam-type conjecture of Baum, Dabrowski, and Hajac (BDH, for short), which states that given a free action of a compact group G on a compact space X, there are no G-equivariant maps $X\ast G\to X$ (with ∗ denoting the topological join). Mainly, we prove the BDH conjecture for locally trivial principal G-bundles. The proof relies on the nonexistence of G-equivariant maps $G^{\ast (n+1)}\to G^{\ast n}$, which in turn is a strengthening of an unpublished result of Bestvina and Edwards. Moreover, we show that the BDH conjecture partially settles a conjecture of Ageev which implies the weak version of the Hilbert–Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold so that the orbit space is finite-dimensional.

We determine the graded decompositions of fusion products of finite-dimensional irreducible representations for simple Lie algebras of rank 2. Moreover, we give generators and relations for these representations and obtain as a consequence that the Schur positivity conjecture holds in this case. The graded Littlewood–Richardson coefficients in the decomposition are parameterized by lattice points in convex polytopes, and an explicit hyperplane description is given in the various types.

Let $\mathtt{V}$ be a standard subspace in the complex Hilbert space $\mathcal{H}$, and let $U:G\to \mathrm{U}(\mathcal{H})$ be a unitary representation of a finite-dimensional Lie group. We assume the existence of an element $h\in \mathfrak{g}$ such that $U(expth)={\mathrm{\Delta }}_{\mathtt{V}}^{it}$ is the modular group of $\mathtt{V}$ and that the modular involution $J_{\mathtt{V}}$ normalizes $U(G)$. We want to determine the semigroup $S_{\mathtt{V}}=\{g\in G:U(g)\mathtt{V}\subseteq \mathtt{V}\}$. In previous work, we have seen that its infinitesimal generators span a Lie algebra on which $\mathrm{ad}h$ defines a 3-grading, and here we completely determine the semigroup $S_{\mathtt{V}}$ under the assumption that $\mathrm{ad}h$ defines a 3-grading on $\mathfrak{g}$. Concretely, we show that the $\mathrm{ad}h$-eigenspaces ${\mathfrak{g}}^{\pm 1}$ contain closed convex cones $C_{\pm }$, such that

$$S_{\mathtt{V}}=exp(C_{+})G_{\mathtt{V}}exp(C_{-}),$$

where $G_{\mathtt{V}}=\{g\in G:U(g)\mathtt{V}=\mathtt{V}\}$ is the stabilizer of $\mathtt{V}$. To obtain this result, we compare several subsemigroups of G specified by the grading and the positive cone $C_U$ of U. In particular, we show that the orbit ${\mathcal{O}}_{\mathtt{V}}=U(G)\mathtt{V}$ with the inclusion order is an ordered symmetric space covering the adjoint orbit ${\mathcal{O}}_h=\mathrm{Ad}(G)h$, endowed with the partial order defined by $C_U$.

Polyhedral products were defined by Bahri, Bendersky, Cohen, and Gitler, to be spaces obtained as unions of certain product spaces indexed by the simplices of an abstract simplicial complex. In this paper, we give a very general homotopy theoretic construction of polyhedral products over arbitrary pointed posets. We show that under certain restrictions on the poset $\mathcal{P}$, which include all known cases, the cohomology of the resulting spaces can be computed as an inverse limit over $\mathcal{P}$ of the cohomology of the building blocks. This motivates the definition of an analogous algebraic construction—the polyhedral tensor product. We show that for a large family of posets, the cohomology of the polyhedral product is given by the polyhedral tensor product. We then restrict attention to polyhedral posets, a family of posets that includes face posets of simplicial complexes, and simplicial posets, as well as many others. We define the Stanley–Reisner ring of a polyhedral poset and show that, as in the classical cases, these rings occur as the cohomology of certain polyhedral products over the poset in question. For any pointed poset $\mathcal{P}$, we construct a simplicial poset $s(\mathcal{P})$, and show that if $\mathcal{P}$ is a polyhedral poset, then polyhedral products over $\mathcal{P}$ coincide up to homotopy with the corresponding polyhedral products over $s(\mathcal{P})$.

By the SYZ construction, a mirror pair $(X,\check{X})$ of a complex torus X and a mirror partner $\check{X}$ of the complex torus X is described as the special Lagrangian torus fibrations $X\to B$ and $\check{X}\to B$ on the same base space B. Then, by the SYZ transform, we can construct a simple projectively flat bundle on X from each affine Lagrangian multisection of $\check{X}\to B$ with a unitary local system along it. However, there are ambiguities of the choices of transition functions of it, and this causes difficulties when we try to construct a functor between the symplectic geometric category and the complex geometric category. In this paper, we prove that there exists a bijection between the set of the isomorphism classes of their objects by solving this problem.