139 problems

Let $(\mathbf{Q},\mathbf{W},\mathbf{v})$ be a quiver with potential and dimension vector, with associated gauge group $\mathsf{G}$, representation space $\mathsf{N}$, stability con…

Let $Q$ be a quiver, let $\overline Q$ be its doubled quiver, and write … where $H^{\mathrm{BM}}$ denotes Borel–Moore homology. Let $Y_{MO}^{Q,+}$ denote the positive half of the M…

A Slodowy variety is a resolution of a slice to an adjoint orbit in the nilpotent cone. Nakajima's conjecture. Slodowy varieties can be realized as quiver varieties. This conjectur…

Let $Q$ be a quiver, let $\mathbb{Y}_+$ be the positive half of the Maulik–Okounkov Yangian, let $K$ be the scalar extension used in the paper, and set…

Lusztig's fermionic conjecture. The Poincaré polynomials of quiver varieties should be described in terms of fermionic forms.

The stronger factor-sequence conjecture. Under these hypotheses, $(\dots,X',Y',\dots)$ is also a factor sequence for $\gamma$.

The tableau interpretation conjecture. The coefficient $c_\mu(r)$ equals the number of sequences of tableaux satisfying the conditions specified by the quiver-coefficient construct…

Let $r$ be rank conditions satisfying the paper's rank-condition hypotheses, let $R_{ij}$ be the associated array of rectangles, and choose a fixed tableau $T_{ij}$ on each…

Let $O^\tau$ be the algebra introduced in the paper, and let $\Gamma$ be the finite group defining the associated Nakajima varieties. Consider the space of isomorphism classes of i…

Let $E_0\to E_1\to\dots\to E_n$ be a sequence of vector bundle morphisms over a nonsingular variety $\mathfrak X$, with generic maps and rank conditions $r=\{r_{ij}\}$ defining a q…

Nakajima's conjecture. The equivariant Kirwan map is surjective for all quiver varieties.

Let $c_\mu(r)$ be a cohomological quiver coefficient, and represent the relevant combinatorial data by a tableaux diagram: a sequence of chosen semistandard Young tableaux together…

Maximal integrable module conjecture. There is an isomorphism of ${\bold U}$-modules

Quiver-variety K-theory module conjecture. The $K$-theory of the quiver variety is isomorphic, as a module for the quantized enveloping algebra of affine type, to the maximal integ…

Let $X$ be a nonsingular variety, let $E_0\to E_1\to\dots\to E_n$ be a sequence of vector bundles and bundle maps, and let $r=\{r_{ij}\}$ be occurring rank conditions. The associat…

Let $\mathfrak g$ be the Lie algebra used in the quiver-variety construction, let $\mathbf w^1$ and $\mathbf w^2$ be the two framing data, let $\mathfrak L(\mathbf w^i)$ be the cor…

Let $\mathfrak g$ be of type $ADE$, let $ath]M(\lambda)$ be the universal standard module, and let $V(\lambda)$ denote Kashiwara's module. Tensoring with $\mathbb Q$ removes the in…

Fix a quiver of finite type, take $m=(1,\dots,1)$ and $\lambda=0$, and let $d,v$ be fixed. Write $M_{m,0}(d,v)$ and $M_{0,0}(d,v)$ for the corresponding quiver varieties. Connected…

Let $M_{0,\lambda}(d,v)$ be the quiver variety in question, and set … Let $(L_d)_0$ denote the $0$-weight space of the Kac–Moody algebra associated to the quiver, for the represent…

Let $I'=I\setminus\{p\}$ index the irreducible components $Y_k$ of the exceptional variety $\mathfrak{L}=\mathfrak{L}_m$, and let $\widetilde E_k(m)$ be the characteristic function…

Let $\mathbf{v}$ be a dimension vector, let $\GL_{\mathbf{v}}$ be the corresponding product of general linear groups, and let $\mathbf{N}_{\mathbf{v}}$ be the associated representa…

Let $\mathcal{L}(n,r)$ be the specified closed projective subvariety of the Gieseker space, let $K_{\mathcal{T}}(\mathcal{L}(n,r))$ denote its equivariant $K$-theory, and let the c…

Let $\mathfrak M_{(0,-2i\zeta_{\mathbb{R}})}$ be the indicated quiver-variety moduli space, let $p_0$ be a fixed point of the $\mathbb{C}^{\star}$-action, and let $W^1(p_0)$ be the…

Let $\mathcal{T}_{\mathbf X}^{\rm 4d}$ be the four-dimensional theory associated with the singularity $x_1^2+x_2x_4^4+x_2^3x_3+x_3^5x_4=0$. In the corresponding five-dimensional an…

Let $\mathbf X$ be the singularity $x_1^5+x_2^3+x_2x_3^3+x_3x_4^2=0$, and let $\mathcal{T}_{\mathbf X}^{\rm 4d}$ be its associated four-dimensional theory. Let the Hasse diagram ob…