Let G be a compact connected Lie group, and let $\operatorname {Hom}({\mathbb {Z}}^m,G)$ be the space of pairwise commuting m-tuples in G. We study the problem of which primes $p \operatorname {Hom}({\mathbb {Z}}^m,G)_1$, the connected component of $\operatorname {Hom}({\mathbb {Z}}^m,G)$ containing the element $(1,\ldots ,1)$, has p-torsion in homology. We will prove that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ for $m\ge 2$ has p-torsion in homology if and only if p divides the order of the Weyl group of G for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ and show that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$-rigid modules for type $A$ and $D$.

We give a realization of Nakajima varieties and the action of the Weyl group on them using certain canonical structures of homological algebras and their natural generalization, which we develop in this paper. We consider in detail the case of an affine quiver, where we present a simple homological characterization of Nakajima varieties and its relation to moduli of sheaves on the projective plane.

Let $F$ be a divisor on the blow-up $X$ of ${{\mathbf{P}}^{2}}$ at $r$ general points ${{p}_{1}},...,{{p}_{r}}$ and let $L$ be the total transform of a line on ${{\mathbf{P}}^{2}}$. An approach is presented for reducing the computation of the dimension of the cokernel of the natural map ${{\mu }_{F}}:\Gamma ({{\mathcal{O}}_{_{X}}}(F))\otimes \Gamma ({{\mathcal{O}}_{_{X}}}(L))\to \Gamma ({{\mathcal{O}}_{_{X}}}(F)\otimes {{\mathcal{O}}_{_{X}}}(L))$ to the case that $F$ is ample. As an application, a formula for the dimension of the cokernel of ${{\mu }_{_{F}}}$ is obtained when $r\,=\,7$, completely solving the problem of determining the modules in minimal free resolutions of fat point subschemes ${{m}_{1}}\,{{p}_{1}}\,+\,\cdot \cdot \cdot \,+\,{{m}_{7}}\,{{p}_{7}}\,\subset \,{{\mathbf{P}}^{2}}$. All results hold for an arbitrary algebraically closed ground field $k$.

Crystal base of the level 0 part of the modified quantum affine algebra $\tilde{U}_q (\widehat{{\frak s}{\frak l}_2})_0$ is given by path. Weyl group actions, extremal vectors and crystal structure of all irreducible components are described explicitly.