Équivalence motivique des groupes algébriques semisimples

Clercq, Charles De

doi:10.1112/s0010437x17007369

Cited by 4 publications

(16 citation statements)

References 47 publications

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“…Proof Table shows that the Tits 2‐index only depends on whether

f_{3} false(G false)

is zero and the 3‐index only depends on whether

g_{3} false(G false)

is zero. Combine this with the main result of . □…”

Section: Tits P‐indexes Of Exceptional Groupsmentioning

confidence: 74%

“…Conversely, if the two subgroups are equal, then for each extension

E

k

, either

{prefixres}_{E / k} false(b (G) false)

is zero and both

G_{E}

and

G_{E}^{'}

are quasi‐split, or it is nonzero and both are anisotropic; in this case the isomorphism of the Tits

p

‐indexes over

k

clearly extends to an isomorphism over

E

. Applying the main result of gives the claim. □…”

Section: Tits P‐indexes Of Exceptional Groupsmentioning

confidence: 90%

“…The groups

G

G^{'}

are motivic equivalent if the groups are motivic equivalent mod

p

for every prime

p

(where

f

may depend on

p

). The main result of says, for a prime

p

G

and

G^{'}

are motivic equivalent mod

p

if and only if there is an isomorphism

f

whose base change to each

p

‐ special field

E

containing

k

identifies the distinguished vertices in the Tits index of

G \times E

with those in the Tits index of

G^{'} \times E

. Informally,

G

and

G^{'}

are motivic equivalent mod

p

if and only if

G \times K

and

G^{'} \times K

have the same Tits

p

‐index for every extension

K

k

.…”

Section: Generalitiesmentioning

confidence: 99%

“…The actual list of indexes is identical with the one in ; the core new material here is the proof of existence of such indexes over 2‐special fields given in the next section. We do not discuss motivic equivalence for classical groups here because the cases of special orthogonal groups and groups of type

^{1} A_{n}

were handled in .…”

Section: Tits P‐indexes Of Classical Groupsmentioning

confidence: 99%

“…Let

G

be a semisimple algebraic group over

k

. As shown in , the Tits

p

‐indexes of

G

on all field extensions of

k

— the higher Tits

p

‐ indexes of

G

— determine the motivic equivalence class of

G

modulo

p

. The aim of this article is to determine the values of the Tits

p

‐indexes of the absolutely simple algebraic groups, using as a starting point the known list of possible Tits indexes as in , or .…”

Section: Introductionmentioning

confidence: 99%

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Tits p‐indexes of semisimple algebraic groups

Clercq

Garibaldi

2017

J. London Math. Soc.

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Abstract. The first author has recently shown that semisimple algebraic groups are classified up to motivic equivalence by the local versions of the classical Tits indexes over field extensions, known as Tits p-indexes. We provide in this article the complete description of the values of the Tits p-indexes over fields. From this exhaustive study, we also deduce criteria of motivic equivalence for semisimple groups of many types, hence giving a dictionary between classic algebraic structures, representation theory, cohomological invariants and Chow motives of the twisted flag varieties for those groups.The Tits index (sometimes called Satake diagram) of a semisimple linear algebraic group G over a field k includes as special cases the classical notions of Schur index of a central simple associative algebra and the Witt index of a quadratic form. It is a fundamental invariant of semisimple algebraic groups. However, for the purpose of stating and proving theorems about Chow motives with F p coefficients, one should consider not the Tits index of G, but rather the (Tits) p-index, meaning the Tits index of G L where L is an algebraic extension of k of degree not divisible by p, yet all the finite algebraic extensions of L have degree a power of p. Such an L is called a p-special closure of k in [8, §101.B] and all such fields are isomorphic as k-algebras, so the notion of Tits p-index over k is well defined.Let G be a semisimple algebraic group over k. As shown in [7], the Tits p-indexes of G on all fields extensions of k -the higher Tits p-indexes of G -determine the motivic equivalence class of G modulo p. The aim of this article is to determine the values of the Tits p-indexes of the absolutely simple algebraic groups, using as a starting point the known list of possible Tits indexes as in [45], [43], or [36]. Along the way, we give in some cases criteria for motivic equivalence for semisimple groups in terms of their algebraic and cohomological invariants.Date: 10 novembre 2015.

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“…Proof Table shows that the Tits 2‐index only depends on whether

f_{3} false(G false)

is zero and the 3‐index only depends on whether

g_{3} false(G false)

is zero. Combine this with the main result of . □…”

Section: Tits P‐indexes Of Exceptional Groupsmentioning

confidence: 74%

“…Conversely, if the two subgroups are equal, then for each extension

E

k

, either

{prefixres}_{E / k} false(b (G) false)

is zero and both

G_{E}

and

G_{E}^{'}

are quasi‐split, or it is nonzero and both are anisotropic; in this case the isomorphism of the Tits

p

‐indexes over

k

clearly extends to an isomorphism over

E

. Applying the main result of gives the claim. □…”

Section: Tits P‐indexes Of Exceptional Groupsmentioning

confidence: 90%

“…The groups

G

G^{'}

are motivic equivalent if the groups are motivic equivalent mod

p

for every prime

p

(where

f

may depend on

p

). The main result of says, for a prime

p

G

and

G^{'}

are motivic equivalent mod

p

if and only if there is an isomorphism

f

whose base change to each

p

‐ special field

E

containing

k

identifies the distinguished vertices in the Tits index of

G \times E

with those in the Tits index of

G^{'} \times E

. Informally,

G

and

G^{'}

are motivic equivalent mod

p

if and only if

G \times K

and

G^{'} \times K

have the same Tits

p

‐index for every extension

K

k

.…”

Section: Generalitiesmentioning

confidence: 99%

^{1} A_{n}

were handled in .…”

Section: Tits P‐indexes Of Classical Groupsmentioning

confidence: 99%

“…Let

G

be a semisimple algebraic group over

k

. As shown in , the Tits

p

‐indexes of

G

on all field extensions of

k

— the higher Tits

p

‐ indexes of

G

— determine the motivic equivalence class of

G

modulo

p

. The aim of this article is to determine the values of the Tits

p

‐indexes of the absolutely simple algebraic groups, using as a starting point the known list of possible Tits indexes as in , or .…”

Section: Introductionmentioning

confidence: 99%

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Tits p‐indexes of semisimple algebraic groups

Clercq

Garibaldi

2017

J. London Math. Soc.

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Critical varieties and motivic equivalence for algebras with involution

Clercq¹,

Quéguiner-Mathieu²,

Zhykhovich³

2022

Trans. Amer. Math. Soc.

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Motivic equivalence for algebraic groups was recently introduced by De Clercq [Compos. Math. 153 (2017), pp. 2195–2213], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in F 2 \mathbb {F}_2 , this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic equivalence of the automorphism groups of two involutions can be checked after extending scalars to some index reduction field, which depends on the type of the involutions. In addition, we describe conditions on the base field which guarantee that motivic equivalent involutions actually are isomorphic, extending a result of Hoffmann on quadratic forms.

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Higher Tate traces of Chow motives

De Clercq,

Quéguiner-Mathieu

2024

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We establish the complete classification of Chow motives of projective homogeneous varieties for p-inner semi-simple algebraic groups, with coefficients in ℤ / p ⁢ ℤ {{\mathbb{Z}}/p{\mathbb{Z}}} . Our results involve a new motivic invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. They apply more generally to objects of the Tate subcategory generated by upper motives of irreducible, geometrically split varieties satisfying the nilpotence principle. Using Chernousov–Gille–Merkurjev decompositions and their interpretation through Bialynicki–Birula–Hesselink–Iversen filtrations due to Brosnan, we then generalize the characterization of the motivic equivalence of inner semi-simple groups through the higher Tits p-indexes. We also define the motivic splitting pattern and the motivic splitting towers of a summand of the motive of a projective homogeneous variety, which correspond for quadrics to the classical splitting pattern and Knebusch tower of the underlying quadratic form.

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Équivalence motivique des groupes algébriques semisimples

Cited by 4 publications

References 47 publications

Tits p‐indexes of semisimple algebraic groups

Tits p‐indexes of semisimple algebraic groups

Critical varieties and motivic equivalence for algebras with involution

Higher Tate traces of Chow motives

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