 and I just for giving me this opportunity. So, I am going to talk on polynomial invariant rings in modular invariant theory and this is joint work with Professor Manoj Goumin. So, first I am going to introduce the subject and then I will discuss the motivation for the questions that we are studying and then I will state the main results and if time permits I will say a few words about the proof. So, let K be a field and V is a finite dimensional K vector space and G is a finite group which is a subset of GLV. So, that is V is a finite dimensional linear representation, faithful linear representation of G over K. So, clearly this action of G induces an action on V dual and which induces an action. So, for each of the elements small G belonging to G, it generates a graded K-algebra automorphism on the symmetric algebra of V dual. So, we denote the symmetric algebra by S which will be a polynomial ring in n variables where n is the dimension of G. So, the ring of invariants of G is the subring of S which is the set of sorry which is the set of all polynomials which are fixed for this action. So, all f in S such that G dot f is equals to f for all G in G. So, when K is algebraically closed we can think of this invariant ring as the ring of functions for the quotient variety V mod G. So, one of the very early questions in invariant theory which was asked is what is the structure of the invariant ring. So, in particular how many generators you need to generate it as a K-algebra or and in particular when is it the simplest of the ring that is when is it the polynomial ring. So, we are going to consider this question in this lecture. So, to answer this question first we have to introduce a kind of elements, a kind of action of an element. So, an element G small G in G is called a pseudo-reflection if it fixes a co-dimension one subspace of V. So, therefore, a pseudo-reflection always has one as a eigenvalue and suppose it has another eigenvalue which is lambda. Now, it can be easily shown that this action is not diagonalized. So, that is action of G on V can be thought of as a matrix. So, this matrix is not diagonalizable if and only if this another eigenvalue is also one and this can happen only when order of G is 0 in K that is K has positive characteristic and that characteristic divide the order of the group and in this case G is called a transvection. So, when the order of the group is 0 in K we call it is the modular case that is the it is in the modular case and when the order of G is invertible in K then it is called the non-modular case. So, our question is properly understood in the non-modular case and it is answered by different people such as Sefer, Todd, Sevalier and Seier for different cases and also like different proofs were given. So, when in the non-modular case it is known that the invariant is a polynomial if and only if G is generated by pseudo-reflection thus a proper characterization is given. However, in a modular case such a characterization is not known but there is a partial converse which is due to say that if S G is a polynomial ring then the action is generated by pseudo-reflections. But in modular case even in small dimension like in dimension 4 for the dimension of V is 4 there are examples where V is generated by pseudo-reflection but S G is not a polynomial ring. So, therefore, we need more conditions in modular situation. So, let us consider, let us compare this modular situation with the non-modular cases and so we now introduce a very important homomorphism in the invariant theory which is the trace homomorphism. So, this trace or transfer homomorphism from S to S G is given by F goes to the sum of its orbits and it is a map of S G modules which one can easily see. Now, if order of G is invertible then we can average it out by order of G and which gives a projection of S onto S G. Therefore, S G is a direct summon of S as an S G module and therefore, the image of this trace map is all of the invariant ring. So, Sank and Oila were proved in 1999 that in the modular situation the image of the trace map is a non-zero proper ideal. Of course, it is contained in the homogeneous ideal generated by the homogeneous invariance of positive degree. And then after several computations and some results they have, they have conjectured that when characteristic of K is P and G is a P group then the invariant ring is a polynomial link if and only if the image of the transfer map is a principal ideal. So, it is a proper ideal that they have proved proper homogeneous ideal and they are conjecturing that when. So, we will be in this situation from now on was that characteristic of K is P and G is a P group and in this case they conjectured that the invariant ring is a polynomial link if and only if the image of the transfer homomorphism is a principal ideal. Now, sorry, sorry. So, in the non-modular case we are seeing that the invariant ring is always a direct summon and in the modular case such is not true. In fact, in most of the cases it would not be a direct summon. So, now, Sank and Oila has given a conjecture which gives, which compares the image of the trace homomorphism with the invariant ring being a polynomial link. Now, we will see that this can be also related to the property of the invariant ring being a direct summon and this is a reformulation by Brewer. So, Brewer has proved that in the situation when R 2 S is a integral extension of integral domains and K and L are fraction field of R and S and such that K 2 L is a finite separable extension. So, then for any element of OI we can consider the multiplication map, this multiplication map and the trace of this map we denote by this. So, then Brewer has proved that if R is a direct summand of S that is if and only if a non-zero principal ideal of S gets mapped to, mapped on to a non-zero principal ideal of R by this trace map. And then he puts this theorem into the situation where I is equal to S G and G is a P group and there he can prove that the S G is a direct summand of S if and only if the image of the transfer homomorphism is a principal ideal. So, therefore, in that case he also proves that G is generated by pseudo reflections. So, therefore, the Sank Oila conjecture can be reformulated as that in the situation in this situation that K is P and G is a P group, the S G is a direct summand if and only if S G is a polynomial link. Now, it is known that if S G is a polynomial link then it is a direct summand. So, therefore, we only have to prove that if S G is a direct summand then it is a polynomial link. So, we will concentrate on proving that for some cases. So, here we see that in the non-modular case the S G is always a direct summand and there is a canonical splitting for the. So, in the modular case we are actually considering a weaker condition, we are considering that S G is a direct summand the splitting may not be canonical in this case. So, now I am going to talk about a class of groups which is called Nakajima groups. This is a class of P groups generated by pseudo reflections. I am not going to give the definition because of time constraints. So, Nakajima introduced this class of P groups and he proved that over prime field S G is a polynomial, if G is a Nakajima group then S G is a polynomial link and over prime field if S G is a polynomial link then G is a Nakajima group. But over bigger fields there are groups which are not Nakajima groups and S G is a polynomial link. So, see in the paper where S Hank and Willow has conjectured their this famous conjecture they have proved that for Nakajima groups this holds that the Sankoilow conjecture holds. So, therefore, Nakajima's proof only said that over prime field the polynomial condition implies Nakajima group, but Sankoilow proves that they are conjectured holds for Nakajima groups. And later Brewer proves for Abelian P groups the Sankoilow conjectured holds and recently Brown has proved for dimension 3 the Sankoilow conjectured holds. So, we introduce a class of P groups which contains the Nakajima groups as a proper subset and it also contains many other representations which are used in invariant theory which are not Nakajima groups and it also contains representations which are not generated by pseudo reflections. So, we proved that the Sankoilow Brewer conjecture holds when G is a generalized Nakajima group. So, the conjecture is known for dimension 3, we proved for dimension 4 over prime field and we also proved for all groups of order P cube these conjecture holds. So, now we are going to introduce another invariance in the invariant theory which is the Hilbert ideal. The Hilbert ideal is a S ideal which is that is the ideal in the big polynomial ring and which is generated by all homogeneous invariants of positive degree. So, we proved that for a generalized Nakajima group the Hilbert ideal is a complete intersection ideal. So, if dimension of V is n then the Hilbert ideal is generated by n many poly, n many homogeneous invariants and each of their degree is less than equal to order of G. So, that the Hilbert ideal should be generated by homogeneous polynomials of degree less than equal to order of G. This is a conjecture of Dixon and Kemper we cannot go deeper into the history of the conjecture. However, this conjecture was not known for Nakajima groups and we not known for Nakajima groups also and we have proved it for generalized Nakajima groups. Now, I am going to give a B. So, I have some 2, 3 minutes 3 minutes. So, I am going to give a sketch for the second statement that we have made here that over prime field and dimension V4 then there is then Sanquilla conjecture holds. So, we consider this relative Hilbert ideal. So, W is a subspace of Vg and W perp is the subspace of linear forms on V which vanishes on W and the relative Hilbert ideal of G with respect to W is an S ideal which is generated by all linear forms that vanishes on W, but they are also invariants. So, it is generated by all those elements and it is a S ideal. So, then we use the existence of a of locally nilpotent derivatives which actually when applied to invariants they remain invariant. So, we use the existence of such derivatives to show that this ideal is actually extended from this smaller ring symmetric algebra of W perp. So, this is the main observation using this we prove that when the Vg is big Vg has dimension greater than equal to dimension of K I mean dimension of V minus 2 then the Hilbert ideal is a complete intersection ideal. So, now, in proving all these all the cases were Sankhelau conjecture hold we use this philosophy of Hilbert we say that if Sg is a direct summand and if there are n many polynomials which generate the Hilbert ideal then Sg has a K algebra generated by those invariants. So, therefore, Sg will become a polynomial ring. So, therefore, we actually prove that the Hilbert ideal is a complete intersection ideal. So, for that we show that in characteristic P and G is a P group which is generated by a transfections in I mean we have discussed that in such cases the pseudo reflections will always be transfections we prove that in this case G has a composition series like this such that each of these G i is a transfection group and G i mod G i minus 1 is isomorphic to Z mod P Z and it is generated by the residue class of a transfection and then we consider the relative Hilbert ideal for this consecutive terms for G i inside G i plus 1 like that. So, that I am denoting here by G prime and G. So, we compare these two relative Hilbert ideals and so in this situation when G mod G prime is Z mod P Z and this is generated by a class of pseudo reflections. So, then we show that in prime field and when rank of V is 4 for any such G prime which is a proper group of a G and in this over prime field and rank V is 4 then rank V G prime will be bigger than equal to rank V minus 2 and then we use our previous theorem to prove that previous theorem of this one this theorem that about large V G. So, we use this theorem to prove that H G prime S w is a complete intersection and which implies then we compare this using this comparison we prove that H G S w is also a complete intersection and from this from existing theorem we get that from Nakajima some earlier results of Nakajima you get that Hilbert ideal is a complete intersection. So, this proves the Senkoilov conjecture for this case like dimension of V is 4 and over prime field. So, these are the references. So, the results about the generalized Nakajima groups are from this paper and the other results are from this manuscript.