I am trying to understand explicit description of spinors in dimension $6$. Now one can explain positive spinors in terms of the usual $4$-dimensional representation of $SU(4)\hookrightarrow GL(4).$ Now what's the corresponding representation for the negative spinors? Is this the only $4$-dimensional representation of $SU(4)$ up to isomorphism? What about the dual representation?
2026-03-27 14:21:43.1774621303
Representation of $SU(4)$
263 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GROUP-THEORY
- What is the intersection of the vertices of a face of a simplicial complex?
- Group with order $pq$ has subgroups of order $p$ and $q$
- How to construct a group whose "size" grows between polynomially and exponentially.
- Conjugacy class formula
- $G$ abelian when $Z(G)$ is a proper subset of $G$?
- A group of order 189 is not simple
- Minimal dimension needed for linearization of group action
- For a $G$ a finite subgroup of $\mathbb{GL}_2(\mathbb{R})$ of rank $3$, show that $f^2 = \textrm{Id}$ for all $f \in G$
- subgroups that contain a normal subgroup is also normal
- Could anyone give an **example** that a problem that can be solved by creating a new group?
Related Questions in REPRESENTATION-THEORY
- How does $\operatorname{Ind}^G_H$ behave with respect to $\bigoplus$?
- Minimal dimension needed for linearization of group action
- How do you prove that category of representations of $G_m$ is equivalent to the category of finite dimensional graded vector spaces?
- Assuming unitarity of arbitrary representations in proof of Schur's lemma
- Are representation isomorphisms of permutation representations necessarily permutation matrices?
- idempotent in quiver theory
- Help with a definition in Serre's Linear Representations of Finite Groups
- Are there special advantages in this representation of sl2?
- Properties of symmetric and alternating characters
- Representation theory of $S_3$
Related Questions in SPIN-GEOMETRY
- Spin Bundle and Connection on $R^3$?
- Meaning of notation in John Roe's Elliptic Operators book
- Spinor bundle of a spin Manifold is a Clifford bundle
- Handle attachment and spin$^c$ structures
- Inclusion between spin groups?
- Understanding of Spin(n) and SO(n)
- Spinor chiral transformation by $\psi \to \gamma^5 \psi$
- On Proposition 2.6 Gualtieri Thesis Generalized complex geometry
- "Square root" of a decomposition of a homogeneous polynomial to harmonic and $x^2 q$.
- Reference request for the irreps of the Spin group
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
We know that $\mathrm{SU}(4)\simeq \mathrm{Spin}(6)$. If we identify their root systems, the positive spinor representation of $\mathrm{Spin}(6)$ corresponds to the usual $4$-dimensional representation $V$ of $\mathrm{SU}(4)$, the natural $6$-dimensional representation of $\mathrm{Spin}(6)$ corresponds to the wedge product $\wedge^{2}V$ and the negative spinor representation corresponds to $\wedge^{3}V$. This can be verified easily by checking their highest weight. I think this is an explicit description of the negative spinor. The dual of the positive spinor representation is the negative spin representation.