Questions tagged [gauge-theory]
Gauge theory in physics and mathematics refers to a field theory whose fields include principal bundles with connection.
216 questions
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Hartshorne's Conjectures about Algebraic Bundles?
In 1978 Hartshorne published a list of 26 open problems about algebraic bundles on projective spaces [Hart], proceeding from an Oxford conference organized by Atiyah.
I understand that many of these ...
user100272
13
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0
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372
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Finite dimensional approximation of Donaldson theory
In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...
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4
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The topology of subgroups of gauge groups
I am reading Atiyah and Bott's classic paper "Yang-Mills Equations over Riemann surfaces" and struggling with proposition 2.16 (p. 542)
Let $P$ be a principal $U(n)$-bundle over a compact Riemann ...
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Deformation-Obstruction Theory of YM Instantons
In Donaldson-Kronhiemer Section 4.2.5. (local models of the moduli space of YM instantons) they first get local models of the moduli space $M$ inside the space of all connections modulo gauge $\...
user100272
5
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1
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Gauge group of tangent bundle and diffeomorphism group
I'm not exactly a differential geometer, so I hope this isn't too elementary a question.
From a naive point of view, it seems as if there are two natural group actions on the space of connections on ...
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7
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Extension problem for Seiberg-Witten solutions
Let $X$ be a compact $4$-manifold, possibly with boundary.
Theorem 17.1.2 of Kronheimer-Mrowka's book "Monopoles and Three-Manifolds" states
Let $X' \subset X$ be a codimension-zero submanifold ...
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Are 2d gauge anomalies determined by genus-one data?
Let $G$ be a (finite, say) group and $\alpha \in \mathrm{H}^3(\mathrm{B}G; \mathrm{U}(1))$ a 3-cohomology class. For each oriented 3-manifold $X^3$ equipped with a $G$-bundle $P : X \to \mathrm{B}G$, ...
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Example of ''annihilation'' of Seiberg-Witten Equation solutions
The proof that the Seiberg-Witten invariants of a 4-manifold $X$ with fixed Spin$^c$ structure really are invariant wrt the metric used to define them goes roughly as follows (for simplicity let $b_2^+...
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tertiary characteristic class: integration of the Chern-Simons form
Let $P \to M$ be a trivial principal circle bundle with connection $A$ over a closed 3-manifold $M$. The Chern-Simons 3-form of the connection is defined by $\mathrm{CS}(A) = A \wedge dA$. Suppose ...
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Todd genus of symplectic $4$-manifolds a smooth invariant?
Suppose that $(M_{1},\omega_{1})$ and $(M_{2},\omega_{2})$ are compact symplectic $4$-manifolds, that are (oriented) diffeomorphic. Is it true that the Todd genus ($\frac{1}{12} (c_{1}^{2} + c_{2})(M_{...
- 6,995
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2
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Elliptic operator becomes Fredholm
Let $X$ be a Riemannian $n$-manifold with tubular end $\mathbb R^+\times Y$, where $Y$ is a closed $n-1$-manifold. Suppose $L:L^{p,w}_2(X)\to L^{p,w}(X)$ is the Laplacian operator which is ...
- 1,915
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3
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About Donaldson-Kronheimer's book on four dimensional manifold
Recently, I read Donaldson-Kronheimer's Geometry of Four Manifolds. It seems that the book requires a lot of background. I had a really hard time digesting the content. Do we have other textbooks ...
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A Question about Hermitian Yang-Mills Equations
Let E be a holomorphic bundle over algebra surface X, let $H$ be a Hermitian metric of $E$, recall the Hermitian-Yang-mills equation is $\wedge F_H=\lambda.1$.
Let $H_t$ be Hermitian metrics over $E$ ...
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11
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2
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827
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Is the space of connections modulo gauge equivalence paracompact?
I find this question interesting, but need to get it out of my system: is the space of connections (modulo gauge) on a compact four-manifold paracompact, in the Sobolev topology?
If so, I believe it ...
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On nowhere zero self-dual 2-forms
Let $(X, g_x)$ be a smooth, oriented, Riemannian 4-diemnsional manifold. Let $\Lambda^2$ denote the bundle of 2-forms over $X$. Then the Hodge-star decomposes $\Lambda^2$ into the space of self-dual ...
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Connected components of gauge groups of principal bundles over generalized flag manifolds
Let $G$ be a compact connected Lie group and $P$ a principal $G$-bundle over a finite CW complex $X$. The gauge group $\mathcal{G}(P)$ is defined to be the group of principal bundle automorphisms of $...
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1
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Killing fields for Yang-Mills
Physicists frequently talk about symmetries of a theory, and them being generated by Killing vectors. While this is clear to me in the context of gravity, where a Killing field $\xi$ is defined by $\...
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962
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Monopole Floer Homology vs. Heegaard-Floer theory
I have a (possibly very naive) question: what is the relation between Monopole Floer Homology and Heegaard-Floer theory? (both known and conjectured)
Is there some version of Atiyah-Floer conjecture ...
- 1,981
15
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2
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Every 4-manifold has a $\operatorname{Spin}^c$ Structure
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}$I'm having trouble understanding the proof given in Morgan's The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-...
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508
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Homology Sphere Embedding into $\mathbb R^4$
Let $Y$ be an oriented closed $3$-manifold, with trivial homology group, i.e. integer homological sphere.
Q: If $Y$ can be embedded into $\mathbb R^4$, is there any example, that such a $Y$ admits a ...
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606
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Gottsche Nakajima Yoshioka define a weird slant product
In their article Instanton counting and Donaldson invariants the authors define the slant product for $\beta \in H_i(X)$ (where $X$ is a manifold) as following.
Let $P \to X$ and SO(3) bundle and $M(...
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3
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1
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Computing the Cartan1-form for $Sp(2)$
Context:
I must find the Cartan 1-form for $Sp(2)$ before I start dealing with the natural connection of the Hopf fibration $S^3 \hookrightarrow S^7 \overset{\mathcal P}\to S^4$. To do so, the idea ...
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205
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Analog of Gauss-Bonnet formula for principal bundles over manifolds with boundary
The Gauss-Bonnet formula gives a topological invariant as an integral over a local density on the given manifold. In particular, when there is a boundary, GB formula has to be supplemented by a ...
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1
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337
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partitions into odd parts vs hooks and symplectic contents
Given a partition $\lambda=(\lambda_1\geq\lambda_2\geq\dots)$, denote the conjugate partition by $\lambda'=(\lambda_1'\geq\lambda_2'\geq\dots)$. For example, if $\lambda=(4,2,2)$ then $\lambda'=(3,3,1,...
- 43.2k
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1
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Is a closed basic 2-form on a principal $S^1$ bundle the curvature of a connection?
Suppose one has an $S^1$ principal bundle $p: P\rightarrow M$, and a closed 2-form $F$ on $M$. Then the pullback form $p^*F$ is closed, vanishes on vertical vectors, and is invariant under the action ...
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3
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1
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Transformation between two conventions of Hitchin equation
Recall that for a given Riemann surface $\Sigma$ Hitchin's self-duality equation consists of a complex rank $r$ vector bundle $E$ (with degree 0 for simplicity), a connection $d_A: \Omega^k(\Sigma, E) ...
- 367
3
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About the Weitzenböck Formula for $SL(2,\mathbb{C})$ connection
Suppose $M$ is a compact four manifold and $P$ is an $SU(2)$ bundle, let $\mathfrak{g}$ be the adjoint bundle of $P$, given a connection $A$ on this bundle. Given $\phi\in \Omega^1(\mathbb{g})$, we ...
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1k
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Linking formulas by Euler, Pólya, Nekrasov-Okounkov
Consider the formal product
$$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$
(a) If $z=2$ then on the one hand we get Euler's
$$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$
on the ...
- 43.2k
8
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251
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Exact triangle for monopole Floer homology with $\mathbb{Z}$-coefficient
Let $Y$ be oriented 3 manifold with torus boundary and let $\gamma_{j}$ (j=0,1,2) be three curves on its boundary with $\#(\gamma_{j}\cap \gamma_{j+1})=-1$. We denote by $Y_{j}$ the manifold obtained ...
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3
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1
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333
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Elementary question: Curvature change under Complexified Gauge Transformation
Forgive me for this elementary question.
Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ ...
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7
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1
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Geometric Construct for Integrating Symmetric Tensors?
I'm interested in finding the appropriate geometric construct for the integration of symmetric tensors, analogous to the way differential forms can be integrated over manifolds.
The motivation comes ...
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2
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591
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Flat connections on 3-manifold with boundary
Suppose $Y$ is a 3-manifold and the boundary $\Sigma:=\partial Y$ is non-empty. Let $G$ be a Lie group with trivial center. Let $\overline {\mathcal A}_{flat}(\Sigma)$ and $\overline {\mathcal A}_{...
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1
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297
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Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$
Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map $f:\mathbb{C}P^2\...
- 5,596
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291
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Deformation of the covariant Laplacian
Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$).
Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...
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0
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361
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Is central extension of a group equivalent to a bundle with gauge field?
Let $\tilde G$ be a central extension of a group $G$ by $U(1)$.
One common elegant definition is that there should be a short exact sequence of groups: $0 \to U(1) \to \tilde G \to G \to 0$
However,...
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2
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1
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107
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A question on 2-bundles
In this paper, the authors +John Baez and +Urs Schreiber defined (page 15) "transition functions" for a special kind of 2-bundles (those whose the base space is a ordinary smooth space augmented to a ...
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Mathematics Book on Yang-Mills Equation [duplicate]
I am planning to read two papers - Atiyah-Bott's paper on Yang-Mills equations on Riemann surfaces and Hitchin's Self-Duality equations on Riemann Surface. Can someone please suggest some book where ...
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2
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1
answer
667
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Sobolev Multiplication theorem for Fibre bundles
Let $X$ be a compact, oriented, four dimensional Riemannian manifold and $Q\longrightarrow X$ be a principal $G$-bundle over $X$ for a smooth, compact Lie group $G$. Let $M$ be a smooth, Riemannian ...
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4
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1
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412
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Symmetries of non-Riemannian curvature tensor
The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection.
By construction it is antisymmetric in the first two indices, since roughly ...
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1
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602
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Prove that the holonomies along any two homotopic paths are the same if the curvature of the connection vanishes [closed]
The proof is trivial in the Abelian case by the Stokes' theorem.How to prove it in the non-Abelian case?
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569
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Bochner-Weitzenbock formula for flat bundle Laplacian
Suppose $(M,g)$ is a compact Riemannian manifold and $(E, \nabla, \lambda, B)$ is the following data:
1) $E$ is a complex vector bundle over $M.$
2) $\nabla$ is a flat connection.
3) $B$ is a ...
- 3,020
4
votes
2
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419
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What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?
Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.
Is it true that if $A$ is the connection 1-form of ...
- 729
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1
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468
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Equivariant Harmonic Maps to R-tree and Korevaar-Schoen Convergence
Thank you for spending time on the following question.
I am trying to make an explicit example of Korevaar-Schoen convergence. The problem I am facing is that I cannot find the limit of the harmonic ...
- 703
1
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1
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868
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Triviality of holomorphic vector bundles over contractible Stein manifolds
If I have correctly undrestood,it is a result of the so called Grauert-Oka principle that all holomorphic vector bundles over contractible Stein manifolds are holomorhically trivial.Does any one knows ...
- 1,236
3
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297
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Cohomology of a flat principal connection
Let $M$ be a compact manifold, $G$ a compact Lie group, $P\to M$ a principal $G$-bundle and $A$ a flat principal connection on $P$. Then $(\Omega^\bullet(M;\operatorname{ad}P),d_A)$ forms a cochain ...
4
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2
answers
1k
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Local structure of the quotient of a Lie group action
Suppose $M$ is a smooth manifold and a compact Lie group $G$ acts freely on $M$, then it is well known that $M/G$ has a manifold structure.
Are there any results for the general case? (a) If the ...
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4
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1
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1k
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Uhlenbeck's theorem novelty
This link provides a short introduction to the contributions of Uhlenbeck about regular gauge fixing. However, I feel quite puzzled about it and I do not understand the real novelty apported by this ...
- 2,091
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0
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580
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On the Hitchin fibration
I will refer to Simpson's "Higgs bundles and local systems".
Proposition 1.4:
When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs ...
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1k
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What is the BRST-anti-BRST formalism?
What is the BRST-anti-BRST formalism?
Is the Sp(2) doublet the ghost, antighost pair?
Introductory accounts of this subject seem to be hard to find. I would appreciate a reference for someone who ...
- 3,880
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0
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197
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computation with Hilbert scheme of $n$ points on $\mathbb C^2$ [closed]
How can we show that
$$\sum_{n = 0}^\infty q^n \operatorname{char}_T S^n(\mathbb C[x,y])=
\prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$
where $\operatorname{char}_T V$ denotes the character ...
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