On the $\mathrm{GL}(n)$-module structure of Lie nilpotent associative relatively free algebras

Publication date: 21 September 2022

Abstract: Let

K l e f t l a n g l e X i g h t a n g l e

denote the free associative algebra generated by a set

X = x_{1}, d o t s, x_{n}

over a field

K

of characteristic

0

. Let

I_{p}

, for

p g e q 2

, denote the two-sided ideal in

K l e f t l a n g l e X i g h t a n g l e

generated by all commutators of the form

[u_{1}, d o t s, u_{p}]

, where

u_{1}, d o t s, u_{p} i n K l e f t l a n g l e X i g h t a n g l e

. We discuss the

m a t h r m G L (n, K)

-module structure of the quotient

K l e f t l a n g l e X i g h t a n g l e / I_{p + 1}

for all

p g e q 1

under the standard diagonal action. We give a bound on the values of partitions

l a m b d a

such that the irreducible

m a t h r m G L (n, K)

-module

V_{l a m b d a}

appears in the decomposition of

K l e f t l a n g l e X i g h t a n g l e / I_{p + 1}

as a

m a t h r m G L (n, K)

-module. As an application, we take

K = m a t h b b C

and we consider the algebra of invariants

(m a t h b b C l e f t l a n g l e X i g h t a n g l e / I_{p + 1})^{G}

for

G = m a t h r m S L (n, m a t h b b C)

,

m a t h r m O (n, m a t h b b C)

,

m a t h r m S O (n, m a t h b b C)

, or

m a t h r m S p (2 s, m a t h b b C)

(for

n = 2 s

). By a theorem of Domokos and Drensky,

(m a t h b b C l e f t l a n g l e X i g h t a n g l e / I_{p + 1})^{G}

is finitely generated. We give an upper bound on the degree of generators of

(m a t h b b C l e f t l a n g l e X i g h t a n g l e / I_{p + 1})^{G}

in a minimal generating set. In a similar way, we consider also the algebra of invariants

(m a t h b b C l e f t l a n g l e X i g h t a n g l e / I_{p + 1})^{G}

, where

G = m a t h r m U T (n, m a t h b b C)

, and give an upper bound on the degree of generators in a minimal generating set. These results provide useful information about the invariants in

m a t h b b C l e f t l a n g l e X i g h t a n g l e^{G}

from the point of view of Classical Invariant Theory. In particular, for all

G

as above we give a criterion when a

G

-invariant of

m a t h b b C l e f t l a n g l e X i g h t a n g l e

belongs to

I_{p}

.

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