All Questions
38 questions
4
votes
0
answers
81
views
Classification of nilpotent orbits over local fields (for type ABCD via partitions )
Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent ...
- 6,622
17
votes
0
answers
402
views
Number of $F_p$-matrices ac=ca, bd = db , ad - da = cb - bc is polynomial in p ? ("Manin matrix variety" - normal ? Cohen–Macaulay ? )
Consider four $n\times n$ matrices $a,b,c,d$ over finite field $F_q$ (or $F_p$ for simplicity), such that they satisfy three equations: $ac=ca,bd=db, ad-da=cb-bc $. Thus an affine algebraic manifold ...
- 24.9k
3
votes
0
answers
202
views
What d.o. $\sum_i f_i(z)\partial_z^i$ correspond to subalgebras $M$ in polynoms $C[x_i]$ being Langlands dual to motive of $Spec(M) \to X$?
Briefly: The question is about presenting explicit examples of the construction discussed in the recent MO question "Relation between motives and geometric Langlands" and Will Sawin's asnwer ...
- 24.9k
1
vote
0
answers
118
views
Schur polynomial with integer values
There is a way to characterize for which $x_1,...,x_d$ a Schur polynomial, that can be defined as
$$s_\lambda(x_1,...,x_d)=\sum_{T\in SSYT(\lambda)}x_1^{t_1}...x_d^{t_d}, $$
with the sum running over ...
11
votes
2
answers
977
views
Reference for combinatorics with view towards representation theory/algebraic geometry
I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
- 1,061
3
votes
0
answers
105
views
Structure of fibers of (complex) moment map of hypertoric variety
I am primarily interested in the hypertoric variety $\mathfrak M(\mathcal B_d)$ associated to the braid arrangement.
Any hypertoric variety $X$, say of complex dimension $2n$, comes equipped with an ...
- 71
14
votes
2
answers
571
views
Number of d-Calabi-Yau partitions
This problem arises from algebraic geometry/representation theory, see https://arxiv.org/pdf/1409.0668.pdf (chapter 2).
We call a partition $p=[p_1,...,p_n]$ with $2 \leq p_1 \leq p_2 \leq ... \leq ...
- 26.5k
10
votes
1
answer
299
views
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.
Let $N^+$ denote the space of uni-upper-triangular ...
- 24.2k
7
votes
1
answer
489
views
Combinatorial meaning of Kazhdan-Lusztig-Stanley polynomial
This question is motivated by
Why do combinatorial abstractions of geometric objects behave so well?
The algebraic geometry of Kazhdan-Lusztig-Stanley polynomials
Kazhdan-Lusztig-Stanley polynomials ...
- 5,230
2
votes
0
answers
94
views
Anything similar to cone product formula (for convex polytopes)?
The convex polytope flag vector ring $\mathcal{R}$ satisfies the cone product formula
$$
C(U) C(V) = C(J(U, V)) + DUV
$$
where
$$
J(U, V) = U C(V) + C(U) V - e_1 UV
$$
is the join formula.
Note: ...
- 459
17
votes
4
answers
2k
views
Reference request: Grassmannian and Plucker coordinates in type B, C, D
Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker ...
- 6,201
10
votes
0
answers
436
views
Commuting matrix variety $[A,B]=0$ - can one geometrically explain divisibility of $F_ q$ point count by high powers of $q$?
$\DeclareMathOperator\Comm{Comm}\DeclareMathOperator\Id{Id}$Consider the variety $\Comm$ of commuting matrices $[A,B]=0$ over some field $K$. It is much studied, and interesting for various reasons.
...
- 24.9k
3
votes
0
answers
175
views
Geometric interpretation of homological quantities in Artinian local Gorenstein algebras
By corollary 3.5. of http://www.ams.org/journals/tran/2012-364-09/S0002-9947-2012-05430-4/S0002-9947-2012-05430-4.pdf the classification of local artinian Gorenstein algebras (all algebras here are ...
- 26.5k
7
votes
1
answer
710
views
Bialynicki-Birula Decomposition and moment polytopes/graphs
Let $X$ be a possibly singular projective scheme which admits a torus $T$ action and has finitely many $T$ fixed points and one-dimensional $T-$orbits. There are many such schemes in the Grassmannian/...
- 1,719
2
votes
1
answer
306
views
connectedness of fibers of torus-equivariant moment maps
Given a possibly singular, connected, symplectic algebraic variety with a torus action, every fiber of the moment map admits a torus action. Is each fiber of this moment map connected? Any examples or ...
- 1,719
3
votes
1
answer
1k
views
Quotients of Grassmannians
Let $G=SL_n(\mathbb C)$ and $T$ be a maximal torus. Then the Grassmannians $Gr(r,n)$ and $G(n-r,n)$ are isomorphic. Now for the left action of the torus on each of them can we say that the GIT ...
- 121
4
votes
1
answer
210
views
A map on Grassmannian
Let $G=SL_{2n}$ and let $\sigma:G \to G$ be defined by $\sigma (A)= E(A^t)^{-1}E^{-1}$, where $E=antidiag(1,1, ... ,1,-1,-1,...,-1)$. Then the maximal parabolic associated to the simple root $\...
- 121
2
votes
1
answer
381
views
Counting cosets in the Quotient of Weyl groups
Let $\Sigma=G/P$ be a flag variety of type $A$,i.e. $G=SL_{n+1}$ and is a stabilizer of the partial flag $0 \subset V_{d_1}\subset \cdots\subset V_{d_r}\subset V_{n+1}$ of length $r$. Let $W$ be its ...
- 83
4
votes
0
answers
76
views
Comparing parametrizations of unipotent radical
Let ${G}$ be a simple algebraic group over $\mathbb{C}$ with maximal torus $T$ and set of simple roots $\{\alpha_i\}_{i\in \Delta}$. We then have a Borel supgroup $B=TU$ with unipotent radical $U$. ...
- 2,516
5
votes
1
answer
733
views
To derive or not to derive, that is the question
What are concrete and abstract examples of problems (even whole programmes of inquiry) when one has a choice to use a "derived" theory (e.g., $\infty$-categories, DAG, HAG, $DRep_k(G)$, "higher" ...
- 639
19
votes
2
answers
1k
views
Explicit invariant of tensors nonvanishing on the diagonal
The group $SL_n \times SL_n \times SL_n$ acts naturally on the vector space $\mathbb C^n \otimes \mathbb C^n \otimes \mathbb C^n$ and has a rather large ring of polynomial invariants. The element $$\...
- 148k
103
votes
3
answers
6k
views
Why do combinatorial abstractions of geometric objects behave so well?
This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference.
Here are two examples of the kind of combinatorial abstractions of geometric ...
- 24.2k
3
votes
1
answer
572
views
Dimension of the zero weight space in $V_{2\rho}$
Let $\rho$ be the half sum of the positive roots (also the sum of fundmental weights) for $SL_n(\mathbb C)$. Then $2ρ$ is in the root lattice. Then what is the dimension of the zero weight space for ...
- 31
1
vote
0
answers
207
views
Singular Locus of a Schubert variety
I am trying to compute the singular locus of the schubert variety $X_w$ in $G_{2,7}$ where $w=(4,7) \in I_{2,7}$. Following the notation in the book "The Grassmannian Variety: Geometric and ...
- 43
4
votes
1
answer
783
views
Three dimensional representations of Alternating group
The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...
- 125
3
votes
1
answer
335
views
Maximal Coset representative for the Weyl group of a Parabolic
Let $G=SL_n$ and let $P_i$ be a maximal parabolic corresponding to a simple root say $\alpha_i$. Let $W_{P_i}$ be the Weyl group of $P_i$. Is there an efficient way to compute the longest coset ...
- 43
4
votes
0
answers
189
views
Fibers of torus equivariant moment maps
Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map
$\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be ...
- 1,719
6
votes
1
answer
778
views
Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$
I need some help about the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ E(f):=\sum_{j=1}^{...
- 307
5
votes
0
answers
171
views
Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B
This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...
- 1,719
6
votes
0
answers
161
views
LS paths construction
Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on $W/W_{...
- 61
6
votes
1
answer
506
views
Lattice model for Affine Grassmannians of non type A
There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional $\mathbb{C}-...
- 1,719
2
votes
0
answers
357
views
$G$-equivariant coherent sheaves on Bott$-$Samelson resolutions
Let $G$ be a Lie group and $B$ a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be ...
- 1,719
9
votes
0
answers
247
views
Degree of a cone over the set of rank $r$ $n\times n$ matrices
Let $X_r\subset Mat_{n\times n}(C)$ denote the variety of rank at most $r$ matrices, set $k=n-r$ and assume $n\geq k^2-1$. Consider the cone over $X_r$ with vertex spanned by the first $k^2-1$ entries ...
- 835
13
votes
1
answer
698
views
Counting representations of $k[x,y]$ when $k$ is finite
$\newcommand{\GFq}{\mathbf F_q}$
Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n(...
- 5,654
2
votes
2
answers
533
views
elements in the weyl group
Let W be the Weyl a group of a semisimple simply connected group over C.
Let I={1,...,r} the set of simple roots.
For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
- 3,472
8
votes
0
answers
636
views
Chern Classes of Exterior Products of a vector bundle.
This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of $...
- 327
23
votes
4
answers
4k
views
What information is contained in the Kazhdan-Lusztig polynomials?
The Kazhdan-Lusztig polynomials contain all kinds of representation theoretic (and other kinds of) informations.
For example the character of a simple module over a Lie algebra with Weyl group $W$ ...
7
votes
1
answer
384
views
Why do non-equioriented A<sub>n</sub> quivers have singularities identical to the singularities of Schubert varieties?
In the general case, quiver cycles are of the form of orbit closures of $GL\cdot V_{\vec{r}}$, where $GL= \prod_{i=0}^n GL_{r_i}$ is the possible changes of basis on all of the vector spaces on each ...
