1,721,266 research outputs found
Lannes T-functor and invariants of pointwise stabilizers
No full textLet rho : G --> GL(n,F) be a representation of a finite group G over the field IF. The group G acts on the algebra of polynomial functions F[V] on V via rho and the subalgebra of polynomials invariant under this action is denoted by F[V](G). If U subset of or equal to V = F-n is a linear subspace then the pointwise stabilizer of U is denoted by G(U) = {g is an element of G \g(u) = u For All u is an element of U}. In this note we examine the relation between F[V](G) and F[V](Gu) when F = F-q is a Galois field with q elements using the T-functor introduced by J. Lannes [13]. We show that a wide variety of properties of F[V](G) are inherited by F[V](Gu). For example, among other things: (1) we reprove a result of R. Steinberg [26] and H. Nakajima [17] that F[V](Gu) is a polynomial algebra when IF[V]G is; (2) we show that the Cohen-Macaulay property is inherited by F[V](Gu) from F[V](G); (3) and when F[V](G) is a complete intersection, then so is F[V](Gu). We apply the T-functor to study degree bounds for generators of rings of invariants, show how the T-functor relates to the transfer homomorphism Tr-G:F[V] --> F[V](G), and give an application in the modular case
On a theorem of R. Steinberg on rings of coinvariants
No full textLet rho : G curved right arrow GL(n,F) be a representation of a finite group G over the field F. Denote by F[V] the algebra of polynomial functions on the vector space V = F-n. The group G acts on V and hence also on F[V]. The algebra of coinvariants is F[V](G) = F[V]/h(G), where h(G) subset of F[V] is the ideal generated by all the homogeneous G-invariant forms of strictly positive degree. If the field F has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that F[V](G) is a Poincare duality algebra if and only if G is a pseudoreflection group. In this note we explore the situation for fields of nonzero characteristic. We prove an analogue of Steinberg's theorem for the case n = 2 and give a counterexample in the modular case when n = 4
On a theorem of Barbara Schmid
No full textLet G be a finite group and rho: G --> GL(n; C) a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions C[V](G) on the vector space V = C-n is generated by homogeneous polynomials of degree at most beta, where beta is the largest degree of a generator in a minimal generating set for C[reg(C)(G)](G), and reg(C)(G) is the complex regular representation of G. In this note we give a new proof of this result, and at the same time extend it to fields F whose characteristic p is larger than \G\, the order of the group G
Noether's bound in the invariant theory of finite groups and vector invariants of iterated wreath products of symmetric groups
No full textLet rho : G --> GL(n, F) be a faithful representation of the finite group G over the held F. In 1916, E. Noether proved that for F of characteristic zero the ring of invariants F[V](G) is generated as an algebra by the invariant polynomials of degree at most \G\. This result has been generalized to the case where the characteristic of F is greater than \G\, or when the characteristic of F is prime to the order of G and the group G is solvable. In this note we show how to refine Noether's proof to yield a more general nonmodular result. In particular we prove that Noether's bound holds for the alternating groups in the nonmodular case
Invariants of 2×2-Matrices over Finite Fields
No full textAbstractLet Fq be the finite field with q elements, q=pν, p∈N a prime, and Mat2.2(Fq) the vector space of 2×2-matrices over F. The group GL(2,F) acts on Mat2,2(Fq) by conjugation. In this note, we determine the invariants of this action. In contrast to the case of an infinite field, where the trace and determinant generate the ring of invariants, several new invariants appear in the case of finite fields
A modular analog of a theorem of R. Steinberg on coinvariants of complex pseudoreflection groups
No full textLet rho : G --> GL(n, F) be a representation of a finite group over the field F, V = F-n the corresponding G-module, and F[V] the algebra of polynomial functions on V. The action of G on V extends to F[ V], and F[V](G), respectively F[V](G), denotes the ring of invariants, respectively coinvariants. The theorem of Steinberg referred to in the title says that when F = C, dim(C) (Tot(C[V](G))) = \G\ if and only if G is a complex reflection group. Here Tot(F[V](G)) denotes the direct sum of all the homogeneous components of the graded algebra F[V](G) and \G\ is the order of G. Chevalley's theorem tells us that the ring of invariants of a complex pseudoreflection representation G --> GL(n, C) is polynomial algebra, and the theorem of Shephard and Todd yields the converse. Combining these results gives: dim(F)(Tot(C[V](G)) = \G\ if and only if C[V](G) is a polynomial algebra. The purpose of this note is to show that the two conditions (i) dim(F)(Tot(F[V](G))) = \G\, (ii) F[V](G) is a polynomial algebra are equivalent regardless of the ground field; in particular in the modular case
On alternating invariants and Hilbert ideals
No full textAbstractIn a paper on F-rationality [J. Algebra 176 (1995) 824–860] Donna Glassbrenner showed that over a field of odd characteristic p the Hilbert ideals of the tautological representations of the symmetric group Σn and alternating group An coincide if n≡0,1modp. She asked if this was always the situation in the modular case. We answer this in the affirmative using Macaulay's theory of irreducible ideals in polynomial algebras: a somewhat forgotten bywater of commutative algebra. As a bonus, the method yields applications back to the original question of F-rationality studied in [J. Algebra 176 (1995) 824–860]
On the dimension of coinvariants of permutation representations
No full textLet f(Z) is an element of F[Z] be a univariate, separable polynomial of degree n with roots x(1), ..., x(n) in some algebraic closure F of the ground field F. It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on F(n) by permutation of the coordinates and each P is an element of F[X(1), ..., X(n)](G) defines a relation P-P(x(1), ..., x(n)) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an H-basis of G-invariant relations if an donly if the algebra of coinvariants F[X(1), ..., X(n)](G) has dimension vertical bar G vertical bar over F. To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite group G has dimension vertical bar G vertical bar if and only if G = Sigma(n) acting via the tautological permutation representation
Gale Monson
No full textOral history interview with Gale Monson as interviewed by Jerry French. Also present for the interview are Larry Smith and Pat French.
Mr. Monson briefly discusses parts of his career, people he worked with, and issues he faced. He also shares a few stories about his time with the Service.
Organization: FWS
Name: Gale Monson
Years: 1933-
Program: Refuges
Keywords:History, Employees (USFWS), Personnel, Wildlife refuges, John Gatlin, dredging, Havasu National Wildlife Refuge, Regional Office, J. Clark Salyer, J.N. "Ding" Darling, wild burrows, Luther Goldman,INTERVIEW WITH GALE MONSON
BY JERRY FRENCH DECEMBER 13, 2002
Also present Larry Smith and Pat French
MR. SMITH: I was interested in your comments about the squatters and the
concessionaires along the river there. Primarily at Imperial, I guess it was.
MR. MONSON: Also at Havasu Lake.
MR. SMITH: But when I came, when I moved in to the Regional Office here at
Albuquerque in 1974, why, they were still having problems with some of those things.
They had a large release of water from that dam there in about 1982. We flooded a lot of
them out when they raised the Colorado River water level. So that got rid of some of
them that way.
MR. MONSON: Our chief antagonist on Havasu Lake was the Bureau of Reclamation.
If I ever disliked any organization, it was the Bureau of Reclamation.
MR. SMITH: I’ve got to tell you, my youngest son works for them!
MR. MONSON: They started dredging right through [unintelligible] Swamp. They
knew I didn’t like it, but there was nothing I could do about it. It just so happened that
this dredge they had, which cost them a hundred and fifty thousand to build to clear the
sand out of this Colorado channel; they were just taking the sand out of the channel and
putting it up on a levee. They were destroying wild habitat by the acres. I was at the
Regional Office; I was called in there. While I was there John Gatlin called me in and he
said, “We just heard today that the dredge sank! It’s sitting on the bottom of the river.
And it’s a good thing you’re in here or you’d get the blame!”
MR. FRENCH: The last time I was out there Gale, they were up on the Bill Williams
unit raising Cottonwoods and trying to get rid of some of that Salt Cedar. They were
raising the cottonwoods on the Bill Williams unit and transplanting them; trying to get
them going down on the riverside where you were fighting them forty years ago.
MR. MONSON: The Bill Williams Refuge was part of the Havasu Lake Refuge for
many, many years. After I left up there, they took Havasu Lake out of the refuge and
turned it over BLM. This was a good idea because BLM inherited all of those squatters
and those concessionaires. It so happened that through a lot of politics, mainly politics,
and probably the exchange of some huge amounts of bribe money a man by the name of,
oh, I can’t think of his name now; but he decided he wanted to own all of that country
next to Havasu Lake on the east side in Arizona. He managed to acquire the whole thing.
Now we have there, Lake Havasu City. It is a city of pretty close to a hundred thousand
people, I believe now. When I was there, there was not a thing there; not a thing!
McCullough was the man’s name. There was nothing there. And during the War when I
was there in World War II, I remember going from Parker Dam to Needles and back again
in the boat.
MR. SMITH: The Crisscraft?
MR. MONSON: Yeah, the Crisscraft. I think I saw one other boat out and I think that
was a squatter at Needles Landing. So things have really changed along the river.
MR. FRENCH: I believe that’s right. Mr. McCullough is the man who owns the chain
saws.
MR. MONSON: Chain saws and outboard motors.
MR. FRENCH: From the money he made from that is how he bought the London Bridge
and moved it to Havasu City. That was all…and as you say, Mr. McCullough is the one
who did that.
MR. MONSON: At one time it was on my refuge!
MR. SMITH: My first assignment out of the Albuquerque Regional Office was in 1974
when Mark Nelson sent me out. He had me fly in a private plane out to the Planet
Ranch; part of Bill Williams to kind of do a biological reconnaissance of that area. I was
picked up by the Refuge Manager from Havasu. He had an add-on air conditioner on the
car. It was a hundred and seven degrees and I hadn’t anything in quite a while.
MR. MONSON: One of the problems in the summer was getting our cars air-conditioned.
The Regional Office took the position that it was a waste of money and
pampering the refuge people. So every time a sent in a purchase order for air conditioning
for our vehicles, George Parker would say ‘no’. Finally, I pointed out to him that the
vehicles of the Arizona Game and Fish Department all had air-conditioning in them. The
vehicles of the Bureau of Reclamation all had air-conditioning in them. That finally
convinced him to okay air-conditioning in our cars. That made a lot of difference in life on
the refuges!
MR. SMITH: I guess he needed to have an extensive visit out there during the hot
season!
MR. FRENCH: Well, if you remember, in the old administrative manual there was a map
of the United States and a line. If you were below that line, you could have a white roof
on your car. Do you remember? We had the old green vehicles.
MR. MONSON: That was after my time I guess.
MR. SMITH: Did you say that you had a residence there at the Desert Museum? Did
you live on the grounds there? Presumably, they had air-conditioning.
MR. MONSON: Yes, it was a swamp cooler. Now it’s being used as an office that has
air conditioning.
MR. FRENCH: Gale, one of the things that we are trying to do find people who died on
the job. Do you remember a man named Ike Kindler who possibly drowned out at
Imperial Lake back in about 1946? He was trying to rescue some duck hunters.
MR. MONSON: No, I don’t remember that.
MR. FRENCH: That’s okay. We’re chasing these things down. It’s just one of those
many things. We are trying to backwards and pick these historical things up. When you
were down at Cabeza; do you want to talk a little bit about the burrows that you fought
with?
MR. MONSON: We didn’t have too much of a problem with them on Cabeza. On the
Kofa, they were running out of our ears! We used [unintelligible] until they passed the
Wild Burrow Protection Act; Wild Horse Anti Project. So now it’s against federal law to
shoot a burrow on federal lands. We used to shoot them every time we saw them. I
remember being with Amonson one time and we came upon a bunch of about thirty
burrows. He was a pretty good shot. I’ve forgotten now what size of rifle it was. We
saw those burrows and they just stood there and looked at us. He got out of the car with
his rifle. He had a whole bunch of cartridges handy. In the next five or so minutes he had
knocked down at least half of them. When one of them is killed, they just stand there.
They don’t run off. We got rid of a lot of burrows that way. We let it be known to the
Game and Fish Department people that they were welcome to shoot all of the burrows
they want or that they could see. So that went on until Wild Horse Anti law was pasted.
That put a stop to it. They have to trap them and moved them out. I understand it’s still
a big headache on Imperial.
MR. FRENCH: Yeah, they are still having quite a bit of problems with them. You’ve
mentioned many names that people are very interested in. Would you talk a little bit
about your relationship with J. Clark Salyer? Many people are interested in what J.
Clark was like.
MR. MONSON: Of course, you know he was a human dynamo. He was appointed to
the job by J. N. “Ding” Darling, who was picked out by Franklin Roosevelt to be the first
Chief of the new FWS. Ding was interested in refuges but he had to find somebody that
really would take hold of the job. He knew Salyer from having had contact with him in
Iowa. Salyer was a high school teacher. He made a name for himself by being very active
in waterfowl conservation. Ickes was really the man who picked who picked Salyer. He
brought Salyer to Washington and Ickes took him over and told him that they had to have
these refuges, “You go out and get them.” Salyer took that as a command and I don’t
know how many years he spent out on the field lining up these refuges; buying them and
getting them any way he could. Many of the refuges are there today on account of J.
Clark Salyer. It was too bad that he became blind. It was very sad, of course. When he
relinquished the job as Chief of the Refuge Division it was a sad day for everybody. I’ve
forgotten that name of the man who first succeeded him but he came down from Region 5
in Boston. He was Chief of the branch when I moved to Washington.
MR. SMITH: Didn’t Dick Griffith have that job?
MR. MONSON: No that was a ways later. Fran Gillette had the job most of the time I
was in Washington.
MR. FRENCH: Gale, you had an amazing career. I look down through these notes, and
it’s amazing. I thank you very much for your time this afternoon. If you’ve got some
other things you’d like to add, or Larry if you have some others questions; but I think
you’ve been very succinct in the whole thing. Like you say; from the day you were born,
right on through your whole career. It was just an amazing career!
MR. SMITH: I think you started in 1933, and I was eight years old when you started
your official job!
MR. FRENCH: [To Mr. Smith] And I’ve always thought of you as one of the old
timers! I wasn’t born until he was just about ready to ship over to India. So that kind of
shows the progression of time through this thing!
MR. MONSON: Yes, it’s hard for me to believe that I am 90 and sitting here talking to
you. It really is!
MR. SMITH: I’d be curious to know if there are any other Refuge Managers that are
that are that old, surviving today.
MR. MONSON: Tom Baugh; you both know him don’t you? He told me that I was
probably the oldest ex-Refuge Manager. It got to me, and I got to thinking about it. I told
him that I realized that maybe I was. I remember Luther Goldman. He was Refuge
Manager at …
MR. FRENCH: Laguna Atascosa was one of them.
MR. MONSON: I think he was first hired to manage the Salton Sea Refuge. Then he
went to Bitter Lake, that’s where I first met him, and subsequently to Laguna Atascosa
and Santa Anna. Then he went to Washington. In Washington he was our Photographer
for years.
MR. SMITH: He advised me on buying cameras at one time.
MR. MONSON: He was a very good photographer. His favorite camera was a
Hasselblad. He took a lot of wonderful pictures.
MR. SMITH: He is still living today?
MR. MONSON: He sure is. I heard from him during the early part of this year saying
that his wife, Betty, had died. He was absolutely devastated. I think he is still living in
the same house in College Park. He’s been there for years and years.
MR. SMITH: You’d think he’d be older than you?
MR. MONSON: I think so. I think he outdates me by, oh, I don’t know how many
years. But I know people who retire and loose their interest real quickly. They don’t
give a hang about refuge history or nothing. But I’m not that way. I keep on being
interested in refuges. I am afraid that some people think I’m too nosey, but…
MR. FRENCH: Well, if it wasn’t for people who were concerned, we wouldn’t have any
of them today. But there are people like Larry; Larry continues to fight for them. And
you’ve been retired for about fifteen years now?
MR. MONSON: He’s doing more than I am.
MR. SMITH: I retired in 1984.
MR. FRENCH: Okay, so we’re getting close to eighteen years on you. And you’re still
fighting for them. Well, I will conclude this. And Gale, again, I thank you very much for
your time.
MR. MONSON: You are very welcome
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